With the defending player not being but acting like a goalkeeper, and assuming the ball would have clearly gone in otherwise.
Can the referee award a goal in this case, even if the ball didn't go in?
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You can't award a goal without the ball crossing the line. The correct decision would be a penalty and a red card for that player for denying a clear goal opportunity.
Sending-off offences A player, substitute or substituted player who commits any of the following offences is sent off:
denying the opposing team a goal or an obvious goal-scoring opportunity by a handball offence (except a goalkeeper within their penalty area)
No. A goal is not awarded automatically. The offensive team is awarded a penalty kick.
From the FIFA Laws of the Game:
Law 10 - Determining the Outcome of a Match
A goal is scored when the whole of the ball passes over the goal line, between the goalposts and under the crossbar, provided that no offence has been committed by the team scoring the goal.
In your case, the ball has not passed over the goal line, so no goal has been scored.
Laws 14 and 12 determine what should happen.
Law 14 - Penalty Kick
A penalty kick is awarded if a player commits a direct free kick offence inside their penalty area or off the field as part of play as outlined in Laws 12 and 13.
Law 12 - Fouls and Misconduct
A direct free kick is awarded if a player commits any of the following offences:
• a handball offence (except for the goalkeeper within their penalty area)
Also in Law 12
Handling the ball
It is an offence if a player:
• deliberately touches the ball with their hand/arm, including moving the hand/arm towards the ball
Putting these together, if a player who is not the goalkeeper deliberately plays the ball with their hand/arm in their own penalty area, a penalty kick shall be awarded to the other team.
No, the referee can not award a goal in this case, and the player must be red-carded. Maybe the most famous case of this was the action of Luis Suarez from Uruguay Selection, making exactly what you ask in 2010 world cup vs Ghana (if I remember well). He was expelled, but in the end the penalty was not converted and Uruguay won the game, So Suarez movement was interesting to me. He interchanged a red card for the possibility of the other team fail a goal that, other way, was obvious.
To quote the rules
Where a player denies the opposing team a goal or an obvious goal-scoring opportunity by a deliberate handball offence the player is sent off wherever the offence occurs.
So, that player would get a red card. Assuming it happens inside the penalty area, the other team also gets a penalty kick. It doesn’t sound as if this would happen outside the penalty box, but if it did, that would be a free kick (from a dangerous position, since you say the shot from there would surely have gone in before).