Your trainer probably showed you a solid wrap, but "proper" wrapping has way more variation than most people realize, and it's likely the culprit here. The pinky side soreness is a red flag - that's usually where people either skip wrapping coverage or don't secure it tightly enough. Have your trainer watch you wrap from start to finish and compare it to how they do it. Small differences in how you anchor around the wrist, how many passes you do across the knuckles, and how tight you pull the tape matter a ton. Some trainers wrap heavier across the knuckle box, others distribute it differently. Also check if you're punching the heavy bag correctly - if you're slightly rolling your wrist or not keeping your hand perfectly flat on impact, the wrap won't protect you the way it should. Bad mechanics can make even a good wrap feel useless.
That said, some soreness early on is normal, especially in your first few months. Your hands are literally not used to this yet. But it shouldn't be painful enough to dread bag work. The fact that it's specifically your pinky side and "heavy" bag work suggests either the wrap's not doing its job or your punch form needs tweaking. Both are fixable. Keep training, but get your trainer to diagnose the wrap first - have them compare how tight and structured theirs feels versus yours. Your hands will toughen up too, but you shouldn't be white-knuckling through sessions waiting for that to happen.