The module IO::Prompt::Hooked verifies the answer and reprompts if wrong. It also provides a convenient way for the user to exit the program.

$type hchana.pl
use strict;
use warnings;
use IO::Prompt::Hooked;
srand(999);     # Delete for production
my $correct_ans;
my %opt = (
    escape  => qr/q(uit)?/i,    # Type 'quit' to exit program
    error   => "Sorry you're wrong, have another go.\n\n",
    validate => sub { $_[0] !~ /\D/ and $_[0] == $correct_ans},
);

while (1) {
    my $random_integer1=int(rand(100));
    my $random_integer2=int(rand(100));
    my $sign =  ($random_integer1 > $random_integer2) ? '-' : '+';
    $correct_ans =  $random_integer1
        + (($sign eq '+') ?  $random_integer2 :  -$random_integer2);
    my $msg = "What is $random_integer1 $sign $random_integer2?";
    my $ans = prompt( message => $msg, %opt );
    last if !defined $ans;
    print "Well done.! $ans is correct.\n\n";
}
warn "Program terminated by user\n";

$perl hchana.pl
What is 10 + 63?  73
Well done.! 73 is correct.

What is 80 + 92?  too hard
Sorry you're wrong, have another go.

What is 80 + 92?  22
Sorry you're wrong, have another go.

What is 80 + 92?  172
Well done.! 172 is correct.

What is 77 - 53?  44
Sorry you're wrong, have another go.

What is 77 - 53?  24
Well done.! 24 is correct.

What is 59 - 41?  quit
Program terminated by user

$
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UPDATE: Started this yesterday. Did not notice that Dave had posted a similar solution while I was sleeping. Note that this version does handle sign the same as the OP.