Hogh-Christensen capsized in his first race and he and Ainslie exchanged angry words in the second.
Ainslie, who got his gold medal bid back on track by winning the first of the day's two races, accused his red-headed rival and Dutchman Pieter-Jan Postma of teaming up against him in the second race when they claimed he hit a mark.
Ainslie did a turn of 360 degrees as a penalty because "two against one in the protest room ...(would be impossible to argue against)".
The Briton, aiming to beat the record four Olympic golds of Hogh-Christensen's fellow Dane, 84-year-old Paul Elvstrom, denied he hit the mark.
"There were some crossed words between me and Jonas at the end because he and the Dutch guy teamed up to claim I hit the mark when I didn’t," he said.
“I had to take the penalty turn so that really fired me up to get past them for the finish and if that’s the way those guys want to play it then fine but they probably didn’t want to fire me up for tomorrow.”
Christensen defended his claim. "That's a bad allegation. Two guys saw him hit the mark, he probably did hit the mark."
He added: "The exchange of words were probably unnecesssary but that's the way it is in sport."
Two races remain before the medal race on Sunday.
Britain's pair of Luke Patience and Stuart Bithell took an early lead overall in the men's 470 two person dinghy, taking second and first in the opening races.
Patience described the two races as very different as they moved from the shore side Nothe course into the West course in the middle of Weymouth Bay from fairly shift winds to 'back to normal with risk management the key to staying upright'.
At least three boats capsized in the second race including the South African, Italian and Argentine craft.
Australia Malcolm Page, who won 2008 gold along with Nathan Wilmot, and his skipper Matthew Belcher were frustrated by their starts and lie fifth overall.
On a good day for the Games hosts, Stevie Morrison and Ben Rhodes won both races in the 49er skiff class on to propel them into second place overall.
Leaders Nathan Outteridge and Iain Jensen of Australia had to do a penalty turn after a Croatian protest. Outteridge said the incident followed a "marginal cross with the Croatian team ... making contact with the tiller extension."
Dutch windsurfer Dorian van Rijsselberge continued his impressive surge towards RS:X gold with another first and second place finish to open up an 11-point lead over his nearest challenger Toni Wilhelm of Germany.
Van Rijsselberge was not getting carried away.
"I'm getting close but I'll celebrate victory when I have a medal round my neck," he said.
Poland's Zofia Noceti-Klepacka enhanced her medal prospects in the women's windsurfing, scoring two firsts to leapfrog Israeli world champion Lee-el Korsitz into second place overall.
Spain's Marina Alabau Neira holds a five-point lead after six races.