How can I train my CNN to learn (numerically) smaller values better?
It seems you are working on a regression problem with a strictly positive target and use MAPE as the loss function to train the neural network. This loss function could be the culprit. As illustrated nicely in this answer, minimizing MAPE leads to predictions that are biased towards smaller values. This is a consequence of the asymmetry between errors in the positive and negative directions (negative errors are 100% at worst, while positive errors are unbounded).

A possible alternative is to minimize the mean absolute error computed with the logarithm of the target. It will converge to the logarithm of the target's median.