The Toffees made a flying start to the campaign with four wins and just one defeat in their opening six Barclays Premier League fixtures, prompting plenty of talk about their chances of claiming a top-four finish to qualify for the Champions League.
They have only lost one more game in the nine that have followed, but their progress has stuttered due to seven of the matches ending in draws and they are currently sixth in the table, three points behind fifth-placed West Brom, fourth-placed Tottenham who they play at Goodison Park on Sunday, and third-placed Chelsea.
While 1-1 stalemates against Manchester City and Arsenal in their last two outings have not left Moyes too displeased at all, some of his side's other draws - they have also emerged with a point from clashes with Norwich, Fulham, Liverpool, QPR, Wigan and, earlier in the season, Newcastle - have been hugely frustrating for the Scot.
He is not losing sight of the fact Everton still have 23 league fixtures ahead of them, though.
Asked if he curses some of the draws, Moyes said: "I do, but it is the start of the season.
"It is still quite early - we are not into Christmas time and a lot will change over the next month or two.
"So you don't need to be right up there right away, even though I'd like to be.
"But yes, I do curse one or two of those results we have picked up."
He added: "I think there are good draws and bad draws.
"The good parts of the Man City and Arsenal matches were that in both games, there were periods where we could have won and there were also periods where I was happy with how we defended and made sure we didn't concede.
"But I think there have been other draws before that where we could say we let points slip and should have done a little bit better."
Everton's bright beginning to this season followed a strong finish to last term, in which January signings such as striker Nikica Jelavic - purchased for a reported £5.5million from Rangers - proved key.
Moyes has indicated he does not expect to be spending much money during the forthcoming January transfer window, although he emphasised that, as in the case of Jelavic, funds to buy a player can suddenly become available.
Asked about the Croatia international and if he was anticipating making a couple of tweaks to his squad in January, Moyes said: "No, I don't think so.
"I think we may go into the loan market to see if we can find anything in that area - I think that is probably where we will be looking.
"But then that was the same as what we were doing last year, and then we made a sale with Diniyar Bilyaletdinov going back to Russia (to join Spartak Moscow) and it freed us up a little bit of cash to go and make the signing. So you never can tell."
Moyes has also made it clear he is not worried about the prospect of other clubs coming in for his players.
"I've said many times - if they want to come, they can come," Moyes said.
"It is no problem for them to come and watch from the stands. But they better come with a big cheque book."