my $term = "\"foo\"[MH] OR \"bar\"[MH]";
my @matches = ( $term =~ /\"(.+)?\"\[MH\]/ );
[download]

As a side note you don't have to do double-quotes-quoting-acrobatics: just use single quotes or alternate delimiters (should you need interpolation or "\*" thingies).

my $term = '"foo"[MH] OR "bar"[MH]';
[download]

or

my $term = qq|"foo"[MH] OR "bar"[MH]|;
[download]