Both previous posters make good points. That said, if you want to fix your code rather than using a module, you have another problem. Your code does not handle the rollover of hours (as it does for minutes). It can all be handled with modulo math. Here is a streamlined version of your while loop with the suggested approach to computing the new time:

while (<DATA>){
    my $former_hour = substr($_, 0, 2);
    my $former_mn = substr($_, 2, 2);

    my $new_hour = ($former_hour + int($diff / 60)) % 24;
    my $new_mn = ($former_mn + $diff) % 60;

    print STDOUT "former_time is $former_hour$former_mn\n";
    print STDOUT "new time is $new_hour$new_mn\n";
}
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Note that I have added line feeds to the print statements.