1

Today during Arsenal vs Crystal Palace, Pierre Emerick Aubamayeng's boot came out after a tackle and he couldn't retrieve the ball, but in case if he managed to retrieve the ball, is he allowed to continue to play immediately, without the boot?

2

Yes he can, but only until the ball next goes out of play.

According to IFAB Laws of the Game 2019/20 Law 04 Page 58

A player whose footwear or shinguard is lost accidentally must replace it as soon as possible and no later than when the ball next goes out of play; if before doing so the player plays the ball and/or scores a goal, the goal is awarded.

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