Once your regex succeeds, $1 holds its value until the next time the regex *succeeds*. You want to conditionalize your use of $1 to only when the regex succeeds, not to when $1 is true:

while (my $line = <FH>) {
    next unless $line =~ m/\s([\w ]+? PRIMARY SCHOOL)/;
    push (@schools, $1);
}
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