Something like this?

$seq = 'TGTGTGTGTGTG';;
@tests = qw[ TX TXXX ];;
for my $test ( @tests ) {
    ( my $re = $test ) =~ tr[X][.];
    print "'$1'" while $seq =~ m[(?=($re))]g;
};;
'TG'
'TG'
'TG'
'TG'
'TG'
'TG'
'TGTG'
'TGTG'
'TGTG'
'TGTG'
'TGTG'
[download]

Replacing your 'X's with '.'s gives you a wildcarding ability.

By placing the capture inside a lookahead, the matchpoint will only be advanced by one position after each match, rather than the length of the match. That allows overlapping matches to be found.