How to deal with sources whose authors I don't have good relationship with? I used to have a good relationship with an author, but now it's broken. However, my work is heavily based on the work of that author, and I'm not sure if citing it without informing them is OK here. Would there be any potential trouble from this? What if they explicitly tell me to not cite theirs? I just don't want to make the relationship worse, because they may not want to have any affiliation with me. But this is just my thought, I really don't know what they really think.
If you feel that the work is relevant, you must cite it - not citing would be unethical.

Also, most people are happy if they are cited - regardless of their relationship to the person who cites them. Citing someone does not mean liking someone - even when criticizing someone's work, you have to cite.

And if you can think of no particular reason why they should not want to be cited by you, there probably is no such reason.