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?