You can append the files into the merged document, just like you are appending to the log file:

use strict;
use warnings;

## open the 'merge' for writing
## this way you will only wrote in the files from this merge
## and not keep anything that is already in there
## to keep any pre-existing content, leave as 
# open(my $merge_fh, '>>', $targetfile) || die "Failed to append to ta
+rget: $!";

## always check for success on open/close FH operations
open(my $merge_fh, '>', $targetfile) 
    || die "Failed open \'$targetfile\' for merging: $!";

my @rpts = glob('//testserver/dir/*');

foreach my $x (@rpts) {
    ## open each file to be merged in turn
    open (my $fh, '<', $x) || die "Failed open \'$x\' for reading : $!
+";
    ## read out each line, printing it into the target file
    while (<$fh>){ print $merge_fh $_;}
    ## close up
    close $fh || die "Failed to close \'$x\' : $!";
}
## now all the files have been copied into the target, you are free to
+ close it.
close $merge_fh || die "Failed to close $targetfile after merge : $!";
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Update: Improved code