Maintain a state flag, and do the extra read once at the end of the loop. For example:

open (file.....)
while (<FILE>) {
   my $skip_next = 0;

   if /blah/ {
       $skip_next = 1;
   }

   elsif ( ... something else ... ) {
   }

   elsif ( ... third condition ... ) {
       $skip_next = 1;
   }

   ....

   $_ = <FILE> if $skip_next;
} # while reading through the file
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Revised: If you need to skip a variable number of lines, you could set $skip_next to the number of lines you wish to skip, and use a loop rather than the final assignment.