How to repair subfloor softened by water damage
It may be easier to repair the sub floor from below depending on the space available. From under the floor you would install the plywood repair piece so that it (if necessary) fit between two floor joists. Good idea when you mentioned applying construction adhesive to the new piece. As a final step have some lengths of 2 x pieces ready and fasten them to the floor joists so as to support the newly installed plywood patch. There is a chance for the joists to be made of steel in which case carriage bolts would be substituted for wood fasteners (and pilot holes drilled in the steel).