Many a time, many a man etc. means many times, many men etc. When we use these expressions we intend to explain a situation where the subject is plural. But structurally ‘many’ follows a plural verb whereas ‘many a’ follows a singular verb though both mean more than one.

EXAMPLES:

• Many men have come here with the same request.
  OR
  Many a man has come here with the same request.
  (here also it means more than one man)
• We have been there many times.
  (OR – you can say –  We have been there many a time)
• I have told you many times not to go there.
  (OR – I have told you many a time not to go there)
• Many young girls have approached the director for a chance in the film.
  (OR – Many a young girl has approached the director for a chance in the film)