Normally, if you have a pseudodragon familiar (via the warlock Pact of the Chain), the way it works is that you can skip your action to allow the pseudodragon to attack with its action. This is lower damage, but if it works, then you’ve permanently given a target disadvantage on attacks.
Also, if you use your pseudodragon familiar in combat, there’s a good chance it will get killed (for example if it provokes an attack of opportunity when flying away after using its sting), and then you’ll pay 10gp to resummon it.
What you have here is different: you have a creature that is not your familiar (since it sounds like you “won” it rather than “summoned” it), so it can use its fire breath every turn in combat, in addition to your action. But, on the other hand, if it tries doing that, something is just going to hit it and kill it. Also, the damage is so small that it’s almost not worth rolling dice for.
(Also, a “1-foot cone” isn’t really a thing in 5e. Either it affects a creature in a 5-foot square, or else it doesn’t.)
If this were your familiar, it would be very annoying to your DM, because the ability you’ve added is phrased as “not an attack” and so it would work in addition to your action. In every combat your pseudodragon would attack for four damage and then a monster would spend an action killing it, and then you’d resummon it for the next combat.
Part of the problem is that this gives you a damage boost that you wouldn’t otherwise get from a familiar, but more of the problem is that you have to roll a lot of dice (including, potentially, tracking damage-over-time effects) and it slows the combat down.
About the Sting ability: it is usually not useful except in very specific combats: using Sting to poison an enemy, if the enemy is important enough to be worth spending a turn to try to give it a debuff, but not important enough to have legendary resistance or a high CON save.
About your proposed ability: it is never useful in combat but might be useful outside of combat. You won’t ever want to get your pseudodragon close enough to an enemy to do its thing, because it’ll get killed and the attack damage is negligible. But you might get some utility from having the pseudodragon light things on fire at a distance.
If I were your DM, I’d say this is probably fine, and I’d allow it.
If I were writing it, I’d simplify it, because right now it’s rolling way too many dice. I’d say:
Flame Puff. Melee weapon attack. +4 to hit, reach 5 ft, one target. Hit: 4 (1d4 + 2) fire damage. The pseudodragon can maintain this ability continuously, like a tiny cute blowtorch.
With this writing, the attack mechanics are exactly the same as Sting, but it’s clear that you’re just trading the “might poison the target” ability for a “light unattended objects on fire” ability. Making it an attack also makes it clear that the familiar can’t use this in parallel with your own attack, if it ever gets made into a familiar.