The only difference between printing to the STDOUT and print to a file is you provide a handle to say or print. So that means

say $fh length for split $motif, $string;
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simply changes to

open(my $fh, '>', $qfn)
   or die("Can't create \"$qfn\": $!\n");

say $fh length for split $motif, $string;
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No need to involve a variable, which would be done as follows:

my @lengths = map length, split $motif, $string;
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