I can't confirm this. When I build a city with say 2 essence and cast a spell that gives 2 gold per essence I get 4 gold from that spell. I can see that in income details. When I build scrying pool and increase city essence to 3, on that turn that pool was built, it still says 4 gold. However, next turn it will show 6 gold and city income will increase correctly.

Same thing with meditation spell that gives 1 mana per essence. That one is easiest to test as you get it at the start of the game and new cities give 0 mana without it so any change in mana income from meditation spell is easy to spot. Try it and see if it works for you. Note that you need to wait one more turn after the pool is built to see the change.

You're right about ending the turn, but that only seems to work for Arcane Forge.  For Gentle Rain, it still doesn't update, and needs a recast.