Depending on when you want to set $self->{_not_exposed_reason},

sub not_exposed_reason {
    my ($self, $not_exposed_reason) = @_;
    return
        $self->{_not_exposed_reason} = defined($not_exposed_reason)
            ? $not_exposed_reason
            : "No exposed reason";
}
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or

sub not_exposed_reason {
    my ($self, $not_exposed_reason) = @_;
    if (defined($not_exposed_reason)) {
        return $self->{_not_exposed_reason} = $not_exposed_reason;
    } else {        
        return "No exposed reason";
    }
}
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Note that

defined(X) ? X : Y
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can be written as

X // Y
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since Perl 5.10.