fill the given soap cup all the way

One thing I've learnt from this thread is that this isn't necessarily right. Its size allows for people with hard water, but if your water is soft then by filling it all rhe way you are likely using too much, which can lead to residue on your dishes. For soft water, half filling is apparently often plenty. Which is another reason to use powder not pods.

Powdered dish detergent and running the water until it gets hot before starting the dishwasher seem to make a difference for me

Also run the tap on your sink until the water turns hot, then start the dishwasher.

A majority of the gunk removal happens during the pre-wash cycle. Hot water greatly helps with this.