Just stash the handle in a member variable and when you call get_next_line call readline on the stashed handle.

package MyFile;
use Moo;

has _fh => { is => 'rw' };

sub BUILD {
  my( $self, @args ) = @_;
  $self->_fh( do {
    open( my $fh, q{<}, $self->frobnicate_path() )
      or die qq{Can't open frobnicated path: $!\n};
    $fh;
  } );
  return;
}

sub frobnicate_path {
  my( $self ) = shift;
  return qq{WHATEVER.txt};
}

sub get_next_line {
  my( $self ) = shift;
  return readline( $self->_fh );
}

sub DEMOLISH {
  if( $self->_fh ) {
    close( $self->_fh ) or warn qq{Problem closing frobnicated path: $
+!\n};
  }
}

1;
__END__
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Edit: fixed readline in get_next_line and added frobnicate_path stub.