The naive way to solve this problem would be to check each of file2's lines against each of file1's entries. Obviously that would take far too long when both files have many entries.

A better way is to craft a regex that combines all the entries of file2. Regexp::Assemble does that for you in an easy and efficient way. Once you have your regular expression made, just apply your regex to each line of file1.

use Modern::Perl;
use Regexp::Assemble;


my @searches = qw/tcgtat gctgga/;

my $ra = Regexp::Assemble->new;
$ra->add( "$_.*" ) for @searches;
$ra = $ra->re;

while (<DATA>) {
    chomp;
    s/$ra//;
    say if $_;
}

__DATA__
xxtcgtatccgaggga
cgcgcgggggagg
jjsjjjjsjjjdtcgtat
aaaaaaacccaaan
ggtcgtatffaadda
gggctggalllslllssdkk
[download]

Output:

xx
cgcgcgggggagg
jjsjjjjsjjjd
aaaaaaacccaaan
gg
gg
[download]