You were almost there. You just needed to open the file which the user provided. Since this can also fail I prefer a loop. eg:

#!/usr/bin/env perl 
use strict;
use warnings;

my $file = '/no_such_file.txt';
my $infh;

while (not open $infh, '<', $file) {
    print "Could not open $file: $!. Please provide another:\n";
    chomp ($file = <STDIN>);
}

my $first = <$infh>;
print "First line of $file: $first";
close $infh;
exit;
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Running this we see something like this:

$ ./fo.pl 
Could not open /no_such_file.txt: No such file or directory. Please pr
+ovide another:
foobar
Could not open foobar: No such file or directory. Please provide anoth
+er:
/etc/hosts
First line of /etc/hosts: 127.0.0.1        localhost.localdomain local
+host
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