Shoving (PHB p. 195) states:
Using the Attack action, you can make a special melee attack to shove a creature
Can the additional attack action granted by the haste spell also be used to shove a creature?
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[...] it gains an additional action on each of its turns. That action can be used only to take the Attack (one weapon attack only) [...] action.
Shoving a creature is a:
[...] special melee attack [...]
but it's not stated to be a weapon attack.
Therefore, you cannot use your hasted action to shove a creature.
The spell haste reads:
and it gains an additional action on each of its turns. That action can be used only to take the Attack (one weapon attack only), Dash, Disengage, Hide, or Use an Object action.
The shove is not counted in the rules as "weapon attacks", they are "special melee attacks".
Using the Attack action, you can make a special melee attack to shove a creature, either to knock it prone or push it away from you. If you're able to make multiple attacks with the Attack action, this attack replaces one of them.
And the same is true for grapple.
When you want to grab a creature or wrestle with it, you can use the Attack action to make a special melee attack, a grapple. If you're able to make multiple attacks with the Attack action, this attack replaces one of them.
Generally, rules mean what the common language means. Weapon attacks are attacks made with weapons, arguable meant to do damage to the target of the attack. We can see that in the designers intent is shown in the Sage Advice rule on smiting where it points out an odd exception to what is considered a weapon attack:
The game often makes exceptions to general rules, and this is an important exception: that unarmed strikes count as melee weapon attacks despite not being weapons.
Note that there is nothing in any texts granting an exception to Shove or Grapple that makes them weapon attacks, even if the one performing the shove or grapple uses the blunt edge of a weapon to do it -- they are still a "special melee attack."
Haste allows you to take the Attack action, but you can only make "one weapon attack".
A weapon attack is, as you can imagine, an attack made with a weapon. This is plain English, it's not a special game term.
Sometimes things that are not attacks with weapons are allowed to count as weapon attacks, for example the rules for unarmed attacks say that you can use use a punch (which is not a weapon) to make a weapon attack:
Instead of using a weapon to make a melee weapon attack, you can use an unarmed strike: a punch, kick, head-butt, or similar forceful blow (none of which count as weapons).
An errata also clarifies that while weapon attacks are attacks made with weapons, some things have rules that make them count as weapon attacks:
What does “melee weapon attack” mean: a melee attack with a weapon or an attack with a melee weapon? It means a melee attack with a weapon. Similarly, “ranged weapon attack” means a ranged attack with a weapon. Some attacks count as a melee or ranged weapon attack even if a weapon isn’t involved, as specified in the text of those attacks. For example, an unarmed strike counts as a melee weapon attack, even though the attacker’s body isn’t considered a weapon.
Here’s a bit of wording minutia: we would write “melee weapon attack” if we meant an attack with a melee weapon.
However, Shove isn't a weapon attack and it doesn't have any extra rules stating that it should count as one:
Using the Attack action, you can make a special melee attack to shove a creature, either to knock it prone or push it away from you.
Unfortunately since Shove isn't a weapon attack (it's a "special" attack) we can't use it with our Hasted Attack action.
It does seem a little strange since it's easy to imagine shoving someone with your weapon, but the rules don't state that is possible. If you are wielding a weapon and want to use it specifically to shove someone, talk to your DM, they may allow it!