Although recent etymology has added "-s" to this word, Mayhap is actually a 16th century term meaning perhaps or perchance. It is actually a derivative of the phrase "may happen" and not a portmanteau of "maybe" and "perhaps".
"I really enjoyed our date. Mayhap I will see you again?"
John has had a cold for nearly a week. Mayhap 'tis time to go to the doctor.