This is the 3534th most frequent Turkish word.
puanını
"The score of it" or "its score."
In this sentence, 'puanını' is used in its accusative case to indicate 'his/her score' as the object of the action 'checking'.
Onun puanını kontrol ettim.
I checked his/her score.
Here, 'puanını' denotes possession and refers to 'his/her score' being increased in the context of the competition.
Yarışmada puanını artırdı.
He/She increased his/her score in the competition.
In this usage, 'puanını' signifies the possessive case indicating the score belonging to the contestant as 'the contestant's score.'