"Can your son count? Mine can!"
Translation:Czy twój syn umie liczyć? Mój umie!
The verb "móc" refers to certain circumstances permitting an action, while the verb "umieć" is more for possessing certain skills or abilities necessary for doing something.
so I suppose that "móć" is "to be allowed" and "umieć" is "to be able"... am I right? ;-)
To some degree, yeah. But „móc” is not just about whether you are being allowed or not. It can refer to situation like you being able to get somewhere on time (because of traffic or weather), finding something misplaced, hitting something that moves fast (but you can use either, depending on whether you consider your skill or evasive movements playing a bigger role), health condition, etc.