It is not that simple, dear Ian :-))
Firstly, it can be Печінка and Пичінка, and even Пічинка - the passport officials seem to be always having a sin of distorting the people's real names :-))

I checked the Dictionary of Ukrainian surnames in Canada and a 2-vol. Dictionary of Ukrainian synonyms (it has excellent etymological references). Personally i think that it was more likely 'Pechinka' (with 'e' in Ukrainian, which was distorted as omophone to 'y').

My guess here, the origins of your surname can be within the area of the Ukrainian words: печеня, печене < > печінки (the last vowel stressed). The first two have a general meaning of some meat dish (roast) and the last word have a meaning of 'inner organs (of people)'.

Secondly, if it orginally was Пічинка in Ukrainian, it could come from the word 'піч' meaning 'stove', 'oven' (with real fire and woods in it).

Lastly, I must say that spelling Пичінка is less likely (the occurence of vowel 'и' in the first syllable in ukraine is quite scarce).

The is very briefly what I think about it.