I've made repairs to several spongy decks...
Knock around with a plastic ball pein hammer to locate the extent of the rot and mark it out. Use a skill saw set around 1/4" depth cut and cut out the area. Dig out the rot and vacuum/clean out then replace with divinicell or any composite core and use fiberglass resin, west system epoxy or similar. Check the level and build up with fiberglass then epoxy the original deck skin back in place. You'll have to address the 1/8' cut that you made either by fiberglassing or if your in a hurry, use 5200 to fill the gap.
This is how I've done it and has worked out perfectly.