How can the Start::output method get hold of the CGI::App methods?

Just pass your CGI::App object to Start when you instantiate it.

sub start {
  my ($self) = @_;
  my $start = Start->new( app => $self );
  return $start->output;
}
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And stick it in your Start object.

package Start;

use strict;
use warnings;

sub new {
  my $class = shift;
  my %args = @_;

  my $self = { 
    app => $args{app};
  };
  bless $self, $class;
}
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And, optionally, add an accessor method if you don't want to peek at the hashref directly.

sub app { 
  my $self = shift;
  return $self->{app};
}
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And then access the object.

sub output {
  my ($self) = @_;
  my $query = $self->app->query;
  my $param = $self->app->param( 'my_param' );
  # do stuff
  return $output;
}
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