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";
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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.