Two-time defending champion Tommy Fleetwood made the cut on the number – at three-under par – courtesy a 15-foot birdie putt on the par-five 18th hole.
Hung Chien-Yao of Taiwan grabbed the early lead with a five-under par 66 in the round which was halted by bad weather.
Kuchar birdied six of his last 10 holes in a four-under par 66 at Waialae Country Club in Honolulu, finishing the week with a 22-under par total of 258.
Putnam, who trailed by one stroke at the halfway stage, shot a three-under 67 to get to 16 under overall. He closed with a birdie Saturday to cut Kuchar's lead to just two.
The 29-year-old American has already resumed putting and chipping, posting photos and updates on her Instagram page, and appeared at a promotional event for equipment-maker Callaway last week.
The three-time major winner succeeds Dane Thomas Bjorn and becomes the just the second golfer from the Republic of Ireland to take the reins.
The 30-year-old qualified for the winners-only field by capturing the Safeway Open for his first PGA Tour victory.
Randhawa, who was ranked in the world top 100 between 2004 and 2009, was taken into detention at the Dudhwa Tiger Reserve in Uttar Pradesh state.
The 42-year-old claimed his first tournament victory since 2013 in front of raucous galleries at the Tour Championship.
The Dane said he would get the tattoo if Europe beat the United States in Paris in September, and on Monday retweeted a video showing him having a 17.5 to 10.5 scoreline and trophy tattoo.
The 30-year-old from Los Angeles closed with a four-under-par 68 at Leopard Creek Country Club for a total of 274 and a two-shot victory over Scot David Drysdale.
The 25-year-old closed with a four-under par 68 for a 20-under total of 268 to finish ahead of compatriot Jazz Janewattananond.
Poom, the winner here in 2016, carded a two-under par 70 to hold his position on top of the leaderboard at 16-under par 200.
John Catlin of the US and South Korea's Sungho Lee set the early pace at the Asian Tour's season-ending event with opening rounds of 66.
Molinari became the first Italian to win a major when he clinched the British Open at Carnoustie earlier this year.