Presley, 49, is the only child of Elvis Presley and ex-wife of both Michael Jackson and Nicolas Cage.
Lockwood is challenging the validity of an agreement he signed after marrying Presley in 2006 that would govern how much he is entitled to in a divorce. He had been seeking $40,000 a month in spousal support, but Cathcart said the unemployed musician would have to wait until after a trial on the agreement.
Lockwood's attorney, Jeff Sturman, argued that Presley had not proven she was deeply in debt and had received an average of $5.6 million in payouts from her father's estate, according to her most recent tax returns.
Presley's lawyer, Mark Gross, contended that Presley has not paid $1 million in taxes she owed last year, is deeply in debt on a property in England and didn't receive the big payout from the estate last year that she has received in previous years.
Lockwood "has no money with which to live," Sturman argued. Leaving him without money from Presley "just leaves Mr. Lockwood in this continuing state of poverty."
Lockwood gave up his career and worked for Presley before their breakup, according to court documents.
Gross contended that Lockwood hasn't been looking for work and should apply to be a music teacher or work at Guitar Center.
Presley and Lockwood both appeared in a downtown Los Angeles courtroom for Wednesday's hearing but did not appear to speak to each other. They separated in June after 10 years of marriage.
Their breakup has grown messy in recent weeks, with Presley disclosing her money trouble. She also said in court filings that she has been living with her adult daughter and has been in a treatment facility for undisclosed reasons since moving from Tennessee to California last year.
She contends the children's court proceeding was initiated after she discovered photos and "disturbing" video on her husband's computer. Sturman has said the allegations are "highly sensational" and inaccurate."
Priscilla Presley tried to allay fans' concerns about the legal issues and well-being of the twins in a Facebook post Sunday.
"There is lots of confusion, commotion and concern from all the talk circulating," she wrote. "Let me put this to rest ... the girls have not been in foster care and never will be. The girls have been with me and will be until all this is sorted out."