The Brazilian winger opened the scoring just before the half-hour mark, but we were pegged back by Tom Cannon's goal just before half-time.
An own goal from Cieran Slicker and a strike from Charlies Whitaker gave the hosts the advantage, and although there was a late own goal from Everton's Ryan Astley, it proved to be a consolation.
We managed to control possession during the early stages, crafting more chances than our Merseyside opponents, and we took the lead just before the half-hour mark thanks to Kayky's first goal since joining City.
However, with just two minutes remaining of the first half, the Toffees equalised through Cannon, seeing the sides go in level at the break.
Into the second half, we were unable to capitalise on our possession and Everton took the lead through an unfortunate own goal from Slicker.
And just before the 80th minute, Everton extended their advantage when substitutes Rhys Hughes and Whitaker combined for the latter to score.
City pulled a goal back late on through an own goal from Everton captain Astley, but we couldn't find an equaliser, seeing us leave Merseyside empty-handed.
HOW WE LINED UP
WHAT IT MEANS
Arsenal's draw with Leicester City sees the Gunners extend their advantage over us to two points.
We remain second in the league at the time of writing, but West Ham could overtake us should they beat Liverpool.
With the international break upon us, a number of our players will join up with their respective national teams, with Cole Palmer, Joshua Wilson-Esbrand, James McAtee and Rico Lewis called up to England's Under-21, Under-20 and Under-18s respectively.
The Elite Development Squad will return to an away fixture to West Ham United on Saturday 20 November.
Kick-off will be at 13:00 (UK).