That is not a "solenoid" above the diverter valve it is a pair of micro switches.
So are you saying that things improve when you put ths screwdriver in?
Can yu get long nosed pliers in to lift the pin up further?
Does that improve it and give hot water?
I'm no expert but it could be the pin has clogged up, or diaphragm is worn. you will only know that by taking it apart & looking, on the 9.24 it's not a hard or long job to do as long as you can get the r/h side panel off.
If it is that then a new diaphragm repair kit is a fiver
here (part 17115).
I replaced
mine a few months ago, was easy enough to do.
You may be able to do it without removing the whole diverter valve, although if you do - given it's age new washers may be needed.
Oh hang on, the 9.24 electronic - where the diverter is on the left?
that may be harder. (Is it behind the DHW HE on that model?)
If you only want to replace the whole diverter they are at the same place I linked above and are not expensive. Thats if it does turn out to be that.
Let's hope it's not the hot water Heat Exchanger - that costs oodles.
(If anyone happens to have a 9.24 they are scrapping with a good DHWHE in I could do with one for mine)