The ionization positive mode depends on the type of the ionisation source used. For example, ionization for ESI, if the acid is in its protonated form can be used in the positive mode because the molecule will receive a proton (H +) and will become have a positive charge that will be detected by the analyzer.
Yes, definitely phenolic acids can be present in the positive mode! You will find some methoxy-conjugated phenolic acids there, for instance, as they ionize better at the positive mode. Are there any ones in particular you are interested in?