Elijah Allman's arrest on March 1 was his second in the US state in a matter of days.
Allman, the 49-year-old son of the singer and actress, was also detained on February 27, accused of acting belligerently at a prestigious prep school in Concord.
It is unclear if Allman, of Malibu, California, has any connection to the home in Windham, New Hampshire.
He is being held in the Rockingham County Department of Corrections, Superintendent Jonathan Banville said.
The hearing on Wednesday was continued until an undetermined date after Allman got a lawyer on Wednesday morning.
The lawyer, Sarah Landres, did not respond to a request for comment.
Allman, whose father was the late singer Gregg Allman, is charged with two counts of criminal mischief, one count of burglary and a count of breach of bail for breaking into the home on March 1.
Police said in a report that Allman did not have permission to be at the home and forcibly entered it.
Officials at St Paul's School said Allman last month identified himself as the parent of a prospective student and slipped into the dining hall as some students were leaving the building.
Police responded to reports that he was disturbing people in the building.
He was charged with four misdemeanours in the school incident: two counts of simple assault, criminal trespass and criminal threatening.
Allman was also charged with a violation of disorderly conduct, which is illegal in the state but not considered a crime.
He was released on bail.
Allman did not respond to an email requesting comment and a phone number for him was not working.
In December 2023, Cher filed a petition to become a temporary conservator overseeing her son's money, saying Allman's struggles with his mental health and addiction have left him unable to manage his assets and potentially put his life in danger.
The petition says the superstar performer's son is entitled to regular payments from a trust fund.
But "given his ongoing mental health and substance abuse issues," she is "concerned that any funds distributed to Elijah will be immediately spent on drugs, leaving Elijah with no assets to provide for himself and putting Elijah's life at risk," the petition says.
Los Angeles Superior Court Judge Jessica Uzcategui denied the request, saying she was not convinced that a conservatorship was urgently needed.
Allman was in the courtroom with his lawyers, who acknowledged his previous struggles but argued that he was in a good place, was attending meetings, getting treatment and reconciling with his estranged wife.
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