"Please finish writing the report as soon as possible."
There are different ways of expressing this but in this sentence every word is needed in its place.
请 (please) + 尽快 (as soon as possible, as fast as you can) + 把 (supporting word)+ 报告 (report) + 写 (write) + 完 (finish)。
把 has to be in that place and is a 介词 which helps form phrases.（介词总是同其他的词组合在一起，构成介词短语，做定语、状语和补语。）
"Please finish writing the report as soon as possible." On another note, I think "Please complete the report soonest possible" should also be accepted.
@bobking12 I think its grammatically incorrect to have a modifier between the verb and the object in Chinese. In your answer the modifier 完 is between the verb 写 and the object 报告. For the sentence to be grammatically correct you have to use the 把 structure as they did in the answer.