Mark Bridger, 47, is accused of abducting and murdering April, who vanished while playing near her home in Wales last October in a case that triggered one of Britain's biggest-ever police search operations.
April's body has never been found, despite hundreds of members of the public joining the search of the mountainous area criss-crossed by rivers.
On the first day of evidence in the high-profile trial, a court in northeast Wales heard that traces of blood matching April's DNA were found in Bridger's living room, hallway and bathroom.
Evans said Bridger, a former abattoir slaughterman, had been seen putting April in his car but has never revealed what he did with her body.
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"It's the defendant's case that he admits that he drove her away," she told the court. "He admits that April is dead. He accepts that he killed her or probably killed her.
Bridger told police that he ran over April with his Land Rover, Evans told the court. She said Bridger, who also previously worked as a lifeguard, had not told police what he did with April's body.
"He says to us that he does not know, that he cannot remember," Evans said.
"It's our case that the defendant's actions -- abduction, murder, covering up what he has done -- that his actions were sexually motivated."
Bridger denies charges of abduction, murder and unlawfully disposing of April's body.
"He has played, we say, a cruel game in pretending not to know what he has done to her," Evans said.
Bridger, who was arrested the day after April disappeared, stared straight ahead as he listened to the evidence.
April, who had cerebral palsy, vanished on October 1 while playing on her bike near her home in the market town of Machynlleth, central Wales.
Her parents had let her play out late as a treat for getting a good report from her teachers.
Police officially called off the search this month.