An Arizona man convicted of murder in the 1984 killing of an eight-year-old girl has been executed.

It was the state’s second execution since officials resumed carrying out the death penalty in May following a nearly eight-year hiatus.

Frank Atwood, 66, died by lethal injection at the state prison in Florence for his murder conviction in the killing of Vicki Hoskinson, whose body was found in the desert, Arizona Attorney General Mark Brnovich said in a statement.

Vicki went missing months earlier after leaving her home in Tucson to drop a birthday card in a nearby mailbox.

The US Supreme Court cleared the way for Atwood’s execution on Wednesday morning after rejecting a final appeal by his lawyers. He died at 10:16am, Mr Brnovich said.

Atwood was the second Arizona prisoner to be put to death in less than a month. The execution of Clarence Dixon last month ended Arizona’s halt to executions that was blamed on the difficulty of obtaining lethal injection drugs and criticism that a 2014 execution in the state was botched.

Death penalty opponents worry that Arizona will now start executing a steady stream of prisoners who have languished on death row, but state officials did not immediately respond to requests for comment on their future execution plans.

No other executions have been scheduled so far in Arizona, which has 111 prisoners on death row, including Atwood.

Atwood was accompanied through the process of being prepared for the lethal injection by a priest, witnesses to the execution said at a news conference. He had claimed innocence, but did not address the killing with his last words.

The witnesses said he thanked the priest for “for coming today and shepherding me into the faith” and added: “I pray the Lord will have mercy on all of us and that the Lord will have mercy on me.”

Witnesses also said the process of setting up IVs into Atwood for the injection went smoothly, though medical personnel could not locate a vein for his right arm and put the IV in his right hand, at Atwood’s suggestion.

Judges in recent weeks had rebuffed attempts by Atwood’s lawyers to have the execution delayed.

Atwood’s attorneys also told the Supreme Court in court filings that the aggravating factor that made his crime eligible for the death penalty was invalidly applied.

He was convicted in 1975 in California for lewd conduct with a child under 14 and was convicted of Vicki Lynne’s killing in 1987. Judges have rejected that legal argument in the past.

The lawyers also questioned whether the state officials met a requirement for the lethal injection drug’s expiration date to fall after the execution date.

Prosecutors claimed that Atwood was trying to indefinitely postpone his execution through legal manoeuvres, saying his pain would be be alleviated by propping him with a pillow on the gurney, which has a tilting function.

Authorities have said Atwood kidnapped Vicki, whose remains were discovered in the desert north-west of Tucson nearly seven months after her disappearance. Experts could not determine the cause of death from the remains, according to court records.

Dixon was executed on May 11 for his murder conviction in the 1978 killing of Deana Bowdoin, a 21-year-old Arizona State University student.

Dixon’s execution was the first to be carried out in the state since the July 2014 execution of Joseph Wood, who was given 15 doses of a two-drug combination over nearly two hours.