A 5/32" depression is not that bad actually. Assuming you don't have a problem with the floor joists that need to be stiffened, you have a couple of options. First, I would span the area with a 8 or 10 foot straight edged board to visualize the areas that are depressed. Much like your pic, I'd pencil out the area on the floor so you can see it clearly. Be sure the existing subfloor is securely nailed or screwed down. Add screws if necessary so there is no bounce.
The easiest method would be to use embossing compound. Mix up a good amount and trowel it on the depressed area. Use your long straight edge to screed it, much like you would a concrete slab. Since your depression is only a little over 1/8", it should be very easy to spread that small amount. Then using a 12" trowel, just gently smooth it over maintaining the depth of fill. Embossing compound in this application is best as it fills easy, dries quickly and easy to sand if you feel you need to remove any ridges, etc left behind after it dries.
Once you have completed this task, evaluate the overall flatness. If you still have some small dips, use more compound or opt for a layer of 3/16" plywood. Remember, always use a layer of red rosin paper between wood layers to prevent squeaking.
You may also opt for a higher quality backer for your engineered flooring. There are several different backers available. They range from ultra cheap thin foam to much better thicker, sound deadening composites. If the engineered wood flooring is 3/8" or thicker, the combination of a good backer and the flooring will bridge any very small irregularities still left on the subfloor. Don't worry about small differences, they will not show. Good luck.
