In this Level the pets come down within mesh, (that need a match beside them to clear), plus the level has two electric blocks (mesh machines) that put mesh onto random boxes and pets, that get within their reach!
After the 1st 1 to 3 pets, don't try to save the next ones that appear, keep trying with adjacent blocks or columns to get more pets showing.
Work as many columns as you can to get many pets. Don't rush them to the bottom but let the pets build up.
Try making moves that effect more than one column.
Some will come down near the mesh machine. Just leave them, and work on others.
Use earned hammers when they are available.
The 2nd, 5th (middle) and 7th column are the best to get multiple pets lined up in.
Also, (it could be just coincidence) but more pets seem to drop if you do not take the mesh away from any pets already on the board - it is a waste of moves anyway to free them until you need to because often they are let out anyway as you make matches to drop them down the board.