How to tell my neural network that I care much more about precision than recall?
I think you have two options that don't exclude each other:


Check precision/recall metrics constantly during training, on Validation data. In this way, during experimentation and hyperparameter tuning, you will choose models that are more favourable for precision rather than recall.
Customize your loss function to correct the model in the right way. The problem is that precision, recall, F1 score, they are all non differentiable functions, i.e. they cannot enter a Loss function as they are. For that reason, some tried to approximate these metrics in a differentiable way. One example is the soft-F1 score, as explained in this article. If you read the function code, you will find differentiable approximations of TP, TN, FP, FN's. You can copy those lines to customize your personalized pseudo-precision Loss function.