Time::Piece (a Core module) can tell you what day of the week it is when your cronjob runs your script every day:

use warnings;
use strict;
use Time::Piece qw(localtime);

my $t = localtime;
my $today = $t->day();

if ($today eq 'Mon') {
    # search and move
}
elsif ($today eq 'Fri') {
    # destroy
}
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