Is there any part in the tefillah (pesukey dizimra, birchos shema, ashrey/uva letzion, etc) where one is allowed to drink coffee/water?
That question was answered over at din.org - water is indeed permissible in most circumstances - coffee (without sugar and milk) typically has the same law as water
After davening Shacharis [on Shabbat], one must make Kiddush before drinking water.
However, this applies only to drinking water for the purpose of quenching one’s thirst. If a person is not thirsty, and wishes to drink water for the purpose of clearing his throat, there is no obligation of kiddush even after Shacharis [see also here on MY]
Before Shacharis, there is no obligation of kiddush, and it is permitted to drink water before davening. During pesukei de-zimra, the problem is the obligation of reciting the shehakol blessing, and similar principles to the above will apply.
If the purpose of drinking is only to clear the throat, no berachah (shehakol) needs to be made, and where required one can drink a little during pesukei de-zimra, without concern for a hefsek (interruption). Poskim discuss whether an ‘action’ (ma’aseh) constitutes a hefsek, and the general concensus is that it does not (see Shut Mekor Chayim, Orach Chaim 2, concerning the act of writing).
If, however, one feels thirsty, a shehakol needs to be made, and one should avoid this during pesukei de-zimra.