Can the Artificer cast Enlarge on one of their eldritch cannons to grant the cannon the extra 1d4 damage to its initial damage? Some of us were confused on the official ruling of that spell.
No, since a Tiny Cannon is not a weapon. For Small or larger Cannons, it is up to the DM.
Consider the case of a Tiny cannon held in one hand:
A Small eldritch cannon occupies its space, and a Tiny one can be held in one hand.
The text of the spell says (emphasis mine):
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.
Enlarge. The target's size doubles in all dimensions, and its weight is multiplied by eight. [...] 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.
Hence, if the Artifice that is holding the Tiny Cannon is targeted by the Enlarge/Reduce spell their weapons change in size and gain the additional 1d4 damage. It seems by description that only wielded weapons can get this bonus.
Unfortunately, The Eldritch Cannon is not a weapon:
You've learned how to create a magical cannon. [...] The cannon is a magical object.
Compare the difference in description with actual magic weapons, whose description explicitly state that they are weapons (see for example a Dragon Slayer).
Hence, even if a Tiny Cannon can be held in one hand (by description) it does not qualify for the additional 1d4 damage.
For a Small or larger Cannons, the description of Enlarge/Reduce is lacking details when it comes to the rules about the modified damage of weapons that are not wielded:
You cause a creature or an object you can see within range to grow larger or smaller for the duration. Choose either a creature or an object that is neither worn nor carried. If the target is unwilling, it can make a Constitution saving throw. On a success, the spell has no effect.
See for example the following Q&A: If I cast the Enlarge/Reduce spell on an arrow, what weapon could it count as?
The final decision is hence on the DM.
A DM may rule otherwise.
Since an Eldritch Cannon acts as a weapon, it is used as a weapon, it does damage as a weapon and has the name of a weapon, a DM may reasonably consider it as a weapon.
In the case of a Tiny Cannon held in one hand, considering the Cannon as a weapon grants the additional 1d4 damage.
Yes. Enlarged Eldritch Cannons Gain Bonus Damage.
The eldritch cannon is an object which is a valid target of enlarge.
Enlarge/reduce increases the size of the target's weapons.
The description and naming of both the ballista and flamethrower forms of eldritch cannon indicate they are armed with weapons.
The cannon exhales fire...
Make a ranged spell attack, originating from the cannon...