McIlroy has been tournament host of the Irish Open for the last four years, during which time it has become part of the lucrative Rolex Series and seen the prize fund surge to £5.3m ($7m).
I'm just sort of stuck in neutral this week.
"I think it's nearly to the point where you need to start caring less about whether the putt goes in and just make a good stroke".
"Tee to green has been really good".
"I feel like I've made some improvements over the last couple of days on that and at least I know I'm trending in the right direction heading into the Open Championship".
"It's just a matter of being a little more efficient and taking my opportunities when I give myself them".
Rory McIlroy bemoans a number of missed putts in the first round of the Irish Open at Ballyliffin on Thursday.
Boris Johnson latest minister to quit since snap election
But Davis said a leadership challenge would be the "wrong thing to do" and insisted he believed May was a "good prime minister".
The three-time major victor recovered from a double bogey on the 13th, caused by a lost ball off the tee, with birdies on the 16th and 17th, only to also double-bogey the last in a back nine of 40.
He's on one-under for the tournament, 13 shots behind South African Erik van Rooyen who scorched to a 66 to take a four-shot lead into the final day.
"The mindset was just keep going, I guess", said the 28-year-old, looking to become only the second South African victor of the Irish Open after Bobby Locke in 1938.
It's going to be very condition-dependent again but if we get a day like this, I'll be doing what I did: "Hit driver everywhere and try to hit wedges in and see if I can hole some putts", said Fox.
"Things have been going well in practice, but you don't always get the results, so that was great to knock off a decent finish there in Italy and move forward", he said.
"It wasn't different on the back nine".