You're mistaken about the content of the variables. What's the output of the following:

{
   use Data::Dumper;
   local $Data::Dumper::Useqq = 1;
   print Dumper \@temp;
   print Dumper $_jobcard_basename;
   print Dumper \@found_files;
}
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Maybe your input hasn't been chomped? It behaves as you think it should:

use strict;
use warnings;

my @temp = '/iri/irs-dev2/dairyassoc/job.trigger/RUNNING.sub-D625448-1
+.pdi_chs.xml';
my $_jobcard_basename = 'pdi_chs';

my @found_files = grep (/$_jobcard_basename/,@temp);

print("Found ", scalar(@found_files), " file(s):\n");
print("$_\n") for @found_files;
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Found 1 file(s):
/iri/irs-dev2/dairyassoc/job.trigger/RUNNING.sub-D625448-1.pdi_chs.xml
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By the way /$_jobcard_basename/ should be /\Q$_jobcard_basename\E/ if $_jobcard_basename contains a string rather than a regexp pattern. It happens to not matter for the particular value it has at the moment.