It's complicated and depends on a number of rulings, but you might not get the extra damage.
Instantaneous vs. ongoing effects
Enlarge / Reduce is a complicated spell in part because it has two different kinds of effects. Some of these effects happen at the time of casting and last for the duration of the spell, while others happen at the time of casting but can end before the spell does. Importantly, none of these effects can begin after the casting of the spell.
When the spell is successfully cast on a creature, the creature changes size for the full duration of the spell:
You cause a creature or an object you can see within range to grow larger or smaller for the duration.
All of the equipment of the creature changes size, but only for so long as the creature sustains it:
If the target is a creature, everything it is wearing and carrying changes size with it. Any item dropped by an affected creature returns to normal size at once.
The spell does not say what happens when the enlarged creature picks up a new object, or an object that has returned to normal size after being set down. Guided by the principle of 'spells do only what they say they do', we would expect nothing to happen in either case, that is, the items would remain at their normal sizes and not enlarge (or re-enlarge). From this, we can conclude that the enlargement itself is an effect that happens only at the time of casting, and not an effect produced (and re-produced) over the duration of the spell itself1.
The spell distinguishes between objects and weapons
As quoted above, all objects the creature "is wearing and carrying changes size with it". This is the general effect on items, described at the start of the spell. The specific effect on weapons, one kind of item, is described later, in the sections for the enlarge and reduce effects. There we find that:
The target's weapons also grow to match its new size. While these weapons are enlarged, the target's attacks with them deal 1d4 extra damage.
The increase in damage applies only to 'these weapons', that is, the weapons that were enlarged upon the casting of the spell. Specific over general, we cannot expect that this increase in damage would apply to all objects carried by the enlarged creature - just those objects that were weapons at the time the spell was cast. Something that was on the person of the creature but not a weapon when the spell was cast should not gain an increase in damage, any more than something that was a weapon when the spell was cast but that the creature only later picked up.
Improvised weapons were not weapons when the spell was cast
The PHB section on Improvised Weapons tells us:
Sometimes characters don't have their weapons and have to attack with whatever is at hand. An improvised weapon includes any object you can wield in one or two hands, such as broken glass, a table leg, a frying pan, a wagon wheel, or a dead goblin.
We note from this that "whatever is at hand" is explicitly not "their weapons". If a PC was carrying a frying pan when they were enlarged, the frying pan would be increased in size as a general effect of the spell on items, but it would not increase in damage capacity because at the time it was enlarged, it was not a weapon. It does not become a weapon until it is 'wielded' as an improvised weapon.
Jeremy Crawford makes this point explicitly when he clarifies:
An improvised weapon is, indeed, a weapon, but only the moment it's used as such. A chair/shield/etc [sic] isn't a weapon otherwise.
In a another tweet, Crawford similarly ruled that the Eldritch Knight's Weapon Bond class feature works only on "bona fide" weapons, not improvised ones.
But, it's bigger, right?
It does seem intuitively reasonable that a larger object would do more damage to the things it struck. However, there are two things that contradict this. The simple objection is that improvised weapons all2 do 1d4 damage. There is considerable difference in size and weight between a table leg and a wagon wheel, yet they both do 1d4 as improvised weapons.
The more complicated objection is a reflection on where the increase in damage 'comes from'. Does it come from the spell itself, 'because the spell says so', or is it a naturalistic interpretation of the consequences of being larger3? Some examination reveals that it is the former, not the latter. Again based on the principle that spells do only what they say they do, we note that while an enlarged creature gains advantage on Strength checks, their absolute strength score is not increased. While their height is doubled, their movement rate is not affected. While a goliath or an ogre might be enlarged to the size of a hill giant, doing so would not grant one the ability to throw rocks to 240 feet. Thus it is clear that the effects of the spell are magical effects, not the natural consequences of a magical increase in size. In keeping with this, it follows that the extra d4 of damage is not 'because' the weapons, improvised or not, are physically larger. Rather, the extra d4 of damage is an additional magical effect of the spell, and as such, it affects only those things which are weapons when the spell is cast4.
You do you
As argued above, I don't think RAW / RAI you get the damage boost from an enlarged, improvised weapon. On the other hand, your players may be expecting it - in fact, they may have selected the spell specifically for that. 'We can get past the weapons checkpoint and enter the grand ball unarmed, but once there the barbarian can grab a chair and the sorcerer will enlarge her...' And it does have an intuitively pleasing feel - swatting someone with a giant frying pan feels like it should hurt more. And it is not at all unbalanced. As a comparison against other 2nd-level spells that require Concentration, an extra d4 per round is quite low. Even when cast on someone getting three attacks per round, such as a dual wielder with extra attack, 3d4 per round assuming you hit is just approaching the baseline of damage output for a 2nd level Concentration spell. So as RAF, I would certainly allow it in some circumstances.
1If you, as a DM, are inclined to rule that items initially enlarged by the spell can be set down to resume their normal size, but are re-enlarged when they are picked back up, then for you the spell is asserting some kind of continuous enlargement effect, not a one-and-done. In this case, it would make more sense for the extra d4 damage to apply at the time the improvised weapons became weapons.
2 Excluding ones that are "similar to actual weapons". A frying pan, in my opinion, isn't particularly similar to an actual weapon - and a larger frying pan even less so.
3 As a biologist, I have to rule that the effects of the spell are all magical, rather than the consequences of a larger size, simply because the naturalistic effects of increasing size proportionally would be so detrimental. A creature that was twice as large would need eight times as much air - but would only have four times the surface area of lungs, and would thus struggle for breath. A creature that was twice as large would be four times as strong, but would be attempting to lift eight times the weight as before, both in its own body and its items. It would struggle to lift its weapons and would collapse under the weight of its own armor.
4 Similarly, a monk's fists and feet increase in size and weight when she is enlarged. And yet, only her staff does more damage, not her unarmed strikes. Again, because her hands and feet are not weapons at the time the spell is cast (or subsequently, for that matter).