The present perfect" have finished" denotes present result of past actions.
Note that the words "away"=out
the word"the letter"=all the letters;
the word"for"=for all the letters referred, when we refer to a period of time.
throwing=a gerund in the form of the present participle of a verb=throw,ending in --ing, for two verbs can't be written in a sentence(grammar).
Hence:-I have finished throwing away (all) the letters lee.