From a German grammar book by Norman Paxton I learned that "doch" and "mal" are used as 'particles' in this sentence in this way:
Doch - It intensifies an imperative, often adding a pleading tone.
Mal - Is more often that not better left untranslated; it lends a pleasant informality of tone.
Particles (auch, denn, doch, eben, eigentlich, etwa, ja, mal, noch, nur, schon, wohl etc.) - These words are exceedingly difficult to translate, and often supply the sentence with a tone which in english is communicated purely by the intonation and so exists only in the spoken and not in the written language.
"Colloquial German stands or falls by an ample scattering of dean, doc, ja, mal, school, so etc., without which it sounds bleak and impersonal" (A. E. Hammer)
Thanks for your very interesting post, notcor11. "Doch" is a particularly hard word to put into English. I knew a German in Berlin who used it to mean "to the contrary" if he disagreed with a statement. My first year German book had it as "yet". My Wirtin once said to me "Doch kannst du das nicht machen!" I knew what she meant, but find it hard to express it in English correctly. The closest I can come is "But you cannot do that".
Other interpretations of 'doch' from the same book...
Doch is used to give an affirmative response where a negative one is expected: Das kann ich mir nicht leisten. Doch! - I can't afford that. Yes, you can.
It is often used to contradict or correct the previous utterance: Es macht doch etwas aus. - But it does matter. Er wird doch kommen. - Yes he will come.
It can also add a sense of 'if only': Hättest du es mir doch gesagt! - If only you had told me!
Also, 'denn doch' expresses indignation or protest: Das geht denn doch zu weit! - That's going too far!
I believe the second one can be applied to your sentence. I find the use of Particles along with the correct usage of prepositions challenging, it must be something that comes with time as you build up a certain 'feeling for the language' (das Sprachgefühl).
Mal is a word that refers to a "time". This is specifically in the mathematical sense rather than the temporal sense. It is often used in math as the multiplication operator, but also to literally describe the number of times something has/does/will happen.
In many uses including this one, mal doesn't have a definite translation in English, but simply makes the request seem more cordial, similar to the use of "on" in "Come on over" which seems more friendly that simply stating "Come over".
You might have chosen "again" because of the phrase "noch einmal" which could be taken by its parts to mean "yet one time" or "one time still". The phrase translates more appropriately to "once more" or "again". It is also sometimes shortened to "nochmal"