Suffolk prosecutors on Tuesday asked the judge presiding over a murder trial to revisit a ruling he made and let them present evidence that the defendant beat his girlfriend, the key witness against him.

Before the trial of Daniel Greenspan began in Riverhead last month, state Supreme Court Justice William Condon ruled that prosecutors would not be allowed to let jurors know that Greenspan’s then-girlfriend, Noriella Santos, had accused him of beating her repeatedly.

The ruling, common in many criminal cases, is designed to prevent a jury from convicting him for crimes other than what he’s charged with.

Greenspan, 30, of Manhattan, is charged with second-degree murder. He is accused of having Santos lure a former lover of hers, Michael Sinclair, from Brooklyn to West Babylon in the early hours of Jan. 31, 2009. There, prosecutors say Greenspan shot Sinclair, 32, to death.

Prosecutors have argued that Greenspan believed Sinclair gave genital herpes to Santos, who then gave it to Greenspan.

Santos, 27, was initially charged with second-degree murder as well, but after agreeing to cooperate against Greenspan she pleaded guilty to second-degree attempted robbery and likely will be sentenced to probation.

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During cross-examination of Santos, defense attorney Arthur Aidala of Manhattan has repeatedly accused her of lying to avoid spending the rest of her life in prison.

But Tuesday, outside the jury’s presence, Assistant District Attorney Sheetal Shetty argued that jurors should hear why Santos did Greenspan’s bidding and why she initially did not turn him in.

“This is a murder case in which the only eyewitness is a participant,” Shetty said. “This defendant has beaten Noriella so many times, and put her in the hospital so many times.”

That’s why she’s getting probation, Shetty said. “We believe her because of what she’s gone through at the hands of this defendant.”

Santos said the defense opened the door to the testimony by asking Santos about Greenspan’s treatment of her and calling attention to the fact that her attorney, Michael Dowd of Manhattan, specialized in battered woman defenses.

Condon will hear further argument on the issue Wednesday before Santos resumes testifying.

Earlier Tuesday, Aidala asked Santos why she got back together with Greenspan in fall 2008 after he left her for another woman earlier that year.

“You’ve escaped from Daniel [at that point], right?” Aidala asked, and Santos agreed. But nevertheless, she moved back in with him, only to have herpes and Sinclair dominate their relationship.

Aidala walked Santos through the many lies she told to Sinclair the night of Jan. 30, 2009, to lure him to Long Island. She said Greenspan orchestrated much of the conversation.

Aidala suggested one lie she told was denying that she killed Sinclair, but she said that was no lie.

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“I didn’t do it, that’s why,” she said.