Can you please post the exact code you are executing and the execution output so I can see the exact warning message? (This is only to satisfy my curiosity; there is no urgency.)

#!/usr/bin/perl -w
use strict;
use warnings;
my %words = map { $_ => 1 } qw($Word $name $test);
my $line = '$Word $ $ $ $test $testing';
$line =~ s/(\$(\w+)*)/exists $words{$1} ? $1 : "dollar$2"/eg;
print "$line\n\n";
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The exact code I am using is this. When I execute this : output is

Use of uninitialized value in concatenation (.) or string at ex.pl lin
+e 6.
Use of uninitialized value in concatenation (.) or string at ex.pl lin
+e 6.
Use of uninitialized value in concatenation (.) or string at ex.pl lin
+e 6.
$Word dollar dollar dollar $test dollartesting
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The problem is with "$2". How to define $2.

Again, the following code, using a regex from postings elsewhere in this thread, always perfectly defines  $2 (although sometimes it is defined as the empty string – perfectly valid!) and seems to give the required output:

>perl -wMstrict -le
"my %words = map { $_ => 1 } qw($Word $name $test);
 my $line = '$Word $ $ $ $test $testing';
 $line =~ s/(\$(\w*))/exists $words{$1} ? $1 : qq{dollar$2}/eg;
 print qq{$line};
"
$Word dollar dollar dollar $test dollartesting
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Note that  (\$(\w*)) is the regex used.