Create a base class for your applications and override run, then inherit from this base class

package MyBase::CGIapp;
use parent qw' CGI::Application ';
sub run {
  my( $self ) = shift;
## pseudo code
  my $ret = eval { $self->SUPER::run(@_); };
  while( $@ ){
     $ret = eval { $self->handle_exeption };
     last if $ret ;
  }

  return $ret;
}
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Run calls all the other methods, cgiapp_prerun....