Moyes started his second spell in charge of West Ham with back-to-back wins as the Hammers overcame Bournemouth and Gillingham in the Premier League and FA Cup respectively.
The capital side suffered a 1-0 defeat at Sheffield United last time out, though, and are still in an extremely vulnerable position in the table during a key stage of the season.
Indeed, just 22 points from 21 matches has left West Ham in 16th spot, just one point clear of the relegation zone. That said, the capital outfit do have a game in hand over 18 of the other teams in the division as their clash with runaway leaders Liverpool had to be postponed.
The Hammers have lost four of their last five Premier League games on home soil, though, while they suffered a 2-0 defeat in the corresponding match with Everton during the 2018-19 campaign.
Everton, meanwhile, will enter Saturday's clash off the back of a 1-0 victory over Brighton & Hove Albion last weekend. The Toffees have won three of their four Premier League games since Ancelotti's arrival, meaning that they have moved clear of the relegation zone after a tough start to the season.
The Merseyside giants are currently 11th in the table, just three points off seventh-placed Wolverhampton Wanderers, which is an indication of just how quickly things can change, even at the top level.
There are obviously still problems for the club's new manager to address, particularly considering that the Toffees suffered a 1-0 loss to a second-string Liverpool in the FA Cup earlier this month.
The likes of Dominic Calvert-Lewin and Richarlison have been in impressive form in recent weeks, though, and the visitors will certainly feel that they can harm West Ham in Saturday afternoon's contest.
As mentioned, Everton ran out 2-0 winners in the corresponding clash last season, while they also recorded a 2-0 victory when the pair locked horns at Goodison Park in October.
West Ham Premier League form: LWLLWL
West Ham form (all competitions): WLLWWL
Everton Premier League form: DDWWLW
Everton form (all competitions): DWWLLW
Darren Randolph is in line to make his second debut for West Ham having completed a return to the Hammers. Indeed, with Lukasz Fabianski unavailable through injury, Moyes confirmed on Thursday that the recent arrival from Middlesbrough was in his thinking for the clash with Everton.
Michail Antonio and Arthur Masuaku are both doubts, though, while Felipe Anderson faces a battle to overcome a rib problem, meaning that Moyes could be forced to make a change in the final third of the field.
As for Everton, Jean-Philippe Gbamin, Alex Iwobi and Andre Gomes are all still on the sidelines, while Ancelotti said on Thursday that central defender Yerry Mina was a doubt having been unable to train this week.
Bernard should again keep his spot in a wide position, though, with Calvert-Lewin and Richarlison continuing their partnership in the final third of the field.
West Ham possible starting lineup:
Randolph; Zabaleta, Ogbonna, Balbuena, Cresswell; Snodgrass, Rice, Noble, Fornals; Lanzini; Haller
Everton possible starting lineup:
Pickford; Coleman, Keane, Holgate, Digne; Sidibe, Davies, Sigurdsson, Bernard; Richarlison, Calvert-Lewin
We say: West Ham 1-1 Everton
This is a tough match to call for a number of reasons. Everton would have fancied their chances of picking up a result in the latter stages of Manuel Pellegrini's tenure, but Moyes does seem to have improved confidence since returning to the London Stadium. We are tipping a low-scoring draw in the capital on Saturday afternoon.