I think this comes from a module that uses (resp. inherits from) Exporter and where Module::Name->import is called with method_name as argument. The simplest case would be:

use Module::Name 'method_name';
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... but that's likely too obvious and you've tried that already. A more contrived way would be something like:

my $module = 'Module::Name';
require 'Module/Name.pm';
$module->import('method_name');
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Maybe (maybe) your code is picking up the wrong versions of Module::Name somewhere else in @INC? Dumping \%INC might tell you which version got picked up.

The easiest "fix" in my opinion would be to preload all the relevant modules at the very top of your program:

#!perl -w
use strict;
use Module::A;
use Module::B;
use Module::Name 'method_name';

...
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That way, all the modules should be loaded in the correct order...