"Pazarcı Kemal elindeki oyun kağıtlarını masanın kirli mermerine atıp kalktı".

Kemal the seller threw the game’s papers to the dirty marble table and got up.

Does my translation make any sense?

And what is the meaning of atlat in "Şu kız gelirse atlat gitsin"?

Yes it makes sense.

atlatmak here means to put (someone) off, get rid of (someone). 

thank youuu

there is a table which has a marble surface... like we use in kitchens but not a kitchen table.