dnd 5e – How does Enlarge affect improvised weapons?

Improvised weapons gain the additional 1d4 damage for a melee attack, but not for a ranged attack.

First of all, the spell’s description states that everything (objects, clothes, weapons…) carried by a creature targeted by Enlarge/Reduce is affected by the spell:

If the target is a creature, everything it is wearing and carrying changes size with it.

In your example, the cooking pot is enlarged as the creature holding it, and it shall return to its normal size once dropped.

A cooking pot is indeed an improvised weapon (it is very similar to the examples given in the rules), and improvised weapons are indeed Weapons: it qualifies hence for the additional 1d4 damage.

The problem is when an enlarged object is used for a ranged attack: as the spell’s description says, once an enlarged item is dropped it returns to its normal size:

Any item dropped by an affected creature returns to normal size at once.

When it is thrown for a ranged attack a strict reading of the rule tells that the pot shrinks and it deals the usual 1d4 damage.

Nonetheless, a DM may rule that since it is considered as a(n improvised) weapon and not as an object, it does not return to its normal size and it deals the additional 1d4 damage even in this case.