$string="Referring Hospital Triage Time <183> 2 h (N <26> 191)"; 
$pre_str="Referring Hospital"; 
$str1=$string;
$str1=~s/\Q$pre_str\E//;

print $str1;
[download]

prints

 Triage Time <183> 2 h (N <26> 191)
[download]

$pre_str=Referring Hospital;
[download]

Perhaps you miss cut-and-paste? But then, if you did, you're showing us different code that you have a problem with. Can you post some runnable code that doesn't do what you expect it to do?

use strict;
my $string="Referring Hospital Triage Time <183> 2 h (N <26> 191)";
my $pre_str="Referring Hospital";
my $str1=$string;
$str1=~s/$pre_str(.*?)/$1/;
print "$str1\n";
[download]

Output: Triage Time <183> 2 h (N <26> 191)

One more thing The \Q tells Perl where to start escaping special characters, and the \E tells it where to stop, and in your $pre_str, i am not finding any special characters in your given example, so no need to use it.

Using regex is kinda like hiring a bulldozer to move a couple shovels-full of dirt. Removing simple, constant strings is a perfect job for index and  substr.

use strict;
my $string="Referring Hospital Triage Time <183> 2 h (N <26> 191)";
my $pre_str="Referring Hospital";

my $index = index $string, $pre_str;
die( "prefix '$pre_str' not present in string '$string'." )
    unless $index >= 0;

my $len = length $pre_str;

my $copy = $string;
substr $copy, $index, $len, '';

print <<"EOMSG"; 
Deleting $len characters starting at postion $index of the string
$string
leaves
$copy
EOMSG
[download]

gemerates the output

Deleting 18 characters starting at postion 0 of the string
Referring Hospital Triage Time <183> 2 h (N <26> 191)
leaves
 Triage Time <183> 2 h (N <26> 191)
[download]

So what do you want to do about that space that isn't part of your prefix string?