It was the second consecutive day that Kuchar shot seven under par at the El Camaleón in Playa del Carmen after his bogey-free opening round on Thursday.
Matt Kuchar is among a trio of players tied for top spot after the first round of the Mayakoba Golf Classic. He was at 12-under 130 and had a one-shot lead among the early starters.
Curtis Luck was the other Australian in the field but the PGA Tour rookie has missed the cut by several shots.
Champ - already a victor on the PGA Tour season - had set the early clubhouse target after firing a round-of-the-day 62, with defending champion Patton Kizzire one of six players a further shot back in tied-third.
Jutanugarn led by four strokes after the second round but slipped to a 71, which included a bogey on the final hole.
He has a 14-under 128 through two rounds.
Kuchar birdied five straight on the front nine and picked up three more birdies coming in to offset his only bogey of the day at the par-four 12th, where his tee shot found the bushes before he limited the damage with a long bogey putt.
Champ said he never thought about the possibility of shooting a 59.
"It's kind of nice to look back and know that I've won on every stage", said Hickcok, who was also a two-time victor on the Mackenzie Tour previous year. "I just had that one challenging hole". The weekend will not include Jordan Spieth.
Seven players are at 11 under, including defending champion Patton Kizzire, past Mayakoba champion Brian Gay and Anirban Lahiri of India, who's played in the last two Presidents Cups, and New Zealand's Danny Lee.
Garcia dominant on day 1 at Nedbank: SUN CITY, South Africa - Sergio Garcia flew into a four-shot lead at the Nedbank Golf Challenge after an 8-under 64 secured the biggest first-round advantage on the European Tour for a year.