This is a standard application for echo cancellation techniques. YOu can make the job easier for echo cancellation by just substracting most of the fixed delay between the two microphones: that allows you to use smaller filters.
Keywords to use a search engine on would be "echo cancellation" and "LMS algorithm".
The basic situation you are stealing the algorithms from is a telephone: you speak into a telephone (which is the good signal) but there also is the signal from the remote speaker which comes out the loudspeaker and ends up in the microphone to some degree. You want to cancel that signal part in the microphone to avoid the remote speaker hearing an echo of his voice.
The difference here is that you don't have an "unadulterated" signal but just a better recording from another microphone. But this should be good enough for engaging the decorrelation algorithms used in echo cancellation, assuming that the speaker influences on the microphones are sufficiently different.