With might and main is 'with all of one's strength'.

Main has been known as meaning 'powerful, possessing physical force' since at least the 12th century. That usage of the word is now virtually lost - apart from as part of might and main. Like chop and change and hard and fast, the phrase is rather tautological as might means near enough the same as main.

There are examples of the use of the phrase, sometimes in the form main and might, in Old English from the late 13th century.