RAW – Absolutely, Large creatures (and Large characters) can wield such weapons with one hand.
The more important questions are
i) whether they will have disadvantage,
ii) whether characters/creatures are proficient with wielding such weapons in one hand, and
iii) what the damage dice will be.
And the answer to those questions are:
i) No, they will not have disadvantage
ii) at the GM’s option, unless proficient with improvised weapons,
iii) d4 or double the dice, depending on ii) above
With Tasha’s Cauldron of Everything introducing the Rune Knight fighter sub-class and it’s 3rd level giant might feature, this question takes on greater significance, as now it applies to a fighter subclass on a daily basis (not just the occasional potion or spell), in addition to Large creatures such as ogres and half-ogres.
Using medium sized two-handed heavy weapons with one Large hand
The rules on improvised weapons provide the basis for this answer. In particular, these rules state that:
Often, an Improvised Weapon is similar to an actual weapon and can be
treated as such. For example, a table leg is akin to a club. At the
GM’s option, a character proficient with a weapon can use a similar
object as if it were that weapon and use his or her Proficiency Bonus. (Weapons, PHB, pg 147)
Consider a maul (Heavy, Two-handed), and something nearly identical in terms of shape and damage type - a warhammer.
Both are basically hammers, dealing bludgeoning damage. They both have long handles, providing plenty of room for grip, regardless of whether it is with two puny paws or one massive fist.
The “medium” (sized for Medium creatures) maul weighs ten pounds, as per the table or martial weapons found on page 149 in the PHB. Presumably a “large” (sized for Large creatures) warhammer weighs eight times as much as a two pound medium Warhammer (weight per the above mentioned martial weapons table), or sixteen pounds in total. The eight times increase is based on extrapolating the guidance in the Enlarge/Reduce spell which – quite logically - indicates that (emphasis mine):
"You cause a creature or an object you can see within range to grow larger or smaller” and further that when an object is enlarged that “The target’s size doubles in all dimensions, and its weight is multiplied by eight. This growth increases its size by one category – from Medium to Large, for example”.
If a Large creature can wield a sixteen pound “large” warhammer in one paw, it can certainly wield a ten pound “medium” maul in one paw. Same with greatclub and club (exact same ratios of ten pounds to two pounds when looking at the martial weapons table in the PHB), and even more so when looking at greataxe/battleaxe (seven pounds to four pounds) or greatsword/longsword (seven pounds to three pounds), or halberd-glaive/spear (six pounds to three pounds) as the math skews even easier for Large creatures when using such weapons.* The only uncertainty being whether the pommel of a “medium” sized greatsword is sufficient for a Large creature’s single hand. Note that the issue is not that the weapon is too large or heavy, rather that part of the weapon is too small. In any event, if a greatsword’s pommel is too small to be one-handed by a Large creature, it is certainly too small to be two-handed by a Large creature.
*A pike/spear (eighteen pounds to three pounds) would be slightly more difficult compared to a maul/warhammer, but it would still be easier to one hand the eighteen pound “medium” pike than the twenty-four pound (three pounds for a medium spear multiplied by eight) large spear.
Mechanically, the rules clearly allow for such weapons to be wielded as improvised weapons, and given that the ("medium"/heavy/two-handed) improvised weapons are lighter than the ("large"/one-handed/versatile) weapons that Large characters (and Large creatures) are proficient in, a GM would be hard pressed to argue that the improvised weapons can’t be wielded one-handed by a Large character or a Large creature (other than the possibility of a greatsword pommel being too small to grasp by a Large hand, but in that case the Large creature wouldn’t be able to wield it at all).
Conclusion to the original question: RAW - Absolutely, creatures and characters that are larger than Medium can use “medium” two-handed weapons with one hand.
As a side note, the Monster Manual on page 238 indicates half-ogres are size Large, just as it shows ogres (on page 237) as size Large, so all of this applies equally to both of them.
Disadvantage with weapons arises from using a weapon that is too large for the character size – small creatures using Heavy medium sized weapons, as per the section on Weapons in the PHB as well as the guidance for monster creation found on page 278 of the DMG. However in this case, it is the opposite: the (medium sized, Heavy, two handed) weapons are slightly lighter than (large sized, one handed/versatile) weapons that characters (and presumably creatures) are proficient in, the weapons are actually sized for a creature smaller than them, not larger. Furthermore, there is no mention of disadvantage in the rules of improvised weapons.
Conclusion to the question of disadvantage: RAW – No, there is no disadvantage to a Large character (or creature) one-handing (medium sized, Heavy) two-handed weapons as improvised weapons.
If a character is proficient with improvised weapons, such as by way of the Tavern Brawler feat, then the proficiency bonus would be applied to attack rolls.
Alternatively, at the GM’s option they can rule that the (“medium”, Heavy, two-handed) improvised weapon is similar to the (“large”, one-handed or versatile) weapon, and if the character (or creature) has proficiency with the “large” weapon, at the GM’s option they can rule that the character (or creature) is proficient with the improvised weapon (the “medium”, Heavy, two-handed weapon) as well, and the proficiency bonus would be applied to attack rolls.
Absent those two conditions, the characters (or creatures) would not be able to apply their proficiency bonus to the attack rolls.
Conclusion to the question of proficiency: RAW – at the GM’s option, unless proficient with improvised weapons.
If the GM has ruled that the weapons are not similar (see proficiency above), then the damage dice would only be 1d4 as for all improvised weapons. Using the (medium) ten pound maul/(large) sixteen pound warhammer as an example, the GM could decide that the nearly 40% weight loss means that the improvised weapon (the “medium” maul) is not a similar weapon to the weapon (the “large” warhammer). Sorry, it’s just too light, and that’s the damage improvised weapons are assigned in the Combat section of the PHB.
Alternatively, the GM could look to the rules on monster creation found in the DMG which state the following:
Big monsters typically wield oversized weapons that deal extra dice of
damage on a hit. Double the weapon dice if the creature is Large,
triple the weapon dice if it’s Huge, and quadruple the weapon dice if
it’s Gargantuan. (DMG pg 278)
In this case the standard 1d4 improvised weapon damage dice would be doubled to 2d4 to reflect that the improvised weapon is twice the size, eight times the weight, and is being wielded by a Large character (or creature) compared to other improvised weapons wielded by Medium size characters. I mean, this improvised weapon is eight times heavier than that improvised weapon, and being used by someone (something) about twice the size and eight times the weight, it’s gonna hurt more.
Finally, if the GM has ruled that the improvised weapon (the “medium” Heavy, two-handed weapon) is similar to the weapon (the “large” one-handed/versatile weapon), then the GM could use those same rules on monster creation, and double the damage dice of a medium sized one-handed/versatile weapon. In the case of a “medium” maul being treated as a “large” warhammer, when wielded one-handed by a Large character (or creature) the damage would be twice the “medium” warhammer, or 2d8. Same for greatsword (1d8 longsword doubled) and greataxe (1d8 battleax doubled). 2d6 for glaive/halberd/pike (1d6 spear doubled). 2d4 for greatclub (1d4 club doubled).
Conclusion to the question of damage dice: RAW – 1d4, 2d4, or 2dX (where X is the damage dice of the similar one-handed/versatile weapon – warhammer d8/longsword d8/battleaxe d8/spear d6/club d4).