According to require, if the argument to require is not a bareword (if it's a variable as you have or an expression returning a name), Perl doesn't automatically add the .pm extension. I suspect you could get by with something like this:

use File::Spec;
my $dir = 'modules';
my $mod = 'test';

my $module = File::Spec->catfile( $dir, $mod . '.pm' );
require $module;
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The documentation also suggests this, which I don't like, though it is shorter: