my @cityKeys;
my @distanceKeys;

foreach( @cgiParamKeys ) {
  if( /city/ ) {  #If a hidden city field found store it in an array;
    push @cityKeys, $_; #Store the field in an array
    @cityKeys = sort @cityKeys;   #Sort the array
  }
  if( /distance/ ) { #If a hidden distance field found, store it in an
+ array
    push @distanceKeys, $_; #Store it in the array
    @distanceKeys = sort @distanceKeys;  #Sort the array
  }
}
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First off, you don't need to sort as you're going - sort it after you've grabbed the items you're interested in. A simpler version might be:

my @cityKeys     = sort grep { /city/     } @cgiParamKeys;
my @distanceKeys = sort grep { /distance/ } @cgiParamKeys;
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That said, I doubt that sorting them is important at all. Next, your "@hiddenKeys". First off, you're creating a double-quoted (qq) string that is all the city and distance keys. That's not what you want. You want my @hiddenKeys = (@cityKeys, @distanceKeys); although you don't even need that. You can just loop through them later:

foreach (@cityKeys, @distanceKeys)
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Inside this loop, you can combine the two conditionals to:

if (/city/ || /distance/)
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But even then, your list of city and distance keys will always match, and you'll always want to output them, so you can just skip the conditional:

foreach  (@cityKeys, @distanceKeys) {
  print $cgi->hidden($_ => $cgiParam{$_});
}
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That said, you could continue to simplify the whole thing by creating a "city hash" that only contained city and distance keys, then you could dump this out even easier. But that's something I'll let you think about after absorbing this far. :-)

(PS: ++ for the progress - good job.)