Definitely "uk". GG5 has an entry for 吐く/つく, but simply points to examples such as 息をつく.
The つく is in some 漢和s, but it's not an approved reading.

http://detail.chiebukuro.yahoo.co.jp/qa/question_detail/q1438771719

I've asked a couple Japanese people about this and they say that they've never 
heard of this kanji read as つく. The つく reading is not part of the Joyo readings 
for this kanji. As far as I can tell, this kanji is usually only pronounced this way 
when it appears in phrases like 「息を吐く」「悪態を吐く」「嘘を吐く」, but is 
usually written in hiragana instead.

@@ -14,0 +15 @@
+<misc>&uk;</misc>
@@ -20,0 +22 @@
+<misc>&uk;</misc>
@@ -27,0 +30 @@
+<misc>&uk;</misc>

they share the vomit sense, but yes, i agree with a split.

@@ -12,4 +12,0 @@
-<r_ele>
-<reb>はく</reb>
-<re_pri>ichi1</re_pri>
-</r_ele>
@@ -17,1 +13,0 @@
-<stagr>つく</stagr>
@@ -20,0 +15,1 @@
+<gloss>to breathe out</gloss>
@@ -21,1 +17,0 @@
-<gloss>to take a breath</gloss>
@@ -24,1 +19,0 @@
-<stagr>つく</stagr>
@@ -27,1 +21,2 @@
-<gloss>to tell (lies)</gloss>
+<gloss>to tell (a lie)</gloss>
+<gloss>to use (foul language)</gloss>
@@ -30,1 +25,0 @@
-<stagr>はく</stagr>
@@ -33,0 +27,1 @@
+<xref type="see" seq="2646460">吐く・はく・1</xref>
@@ -34,1 +29,2 @@
-<gloss>to disgorge</gloss>
+<gloss>to throw up</gloss>
+<gloss>to spit up</gloss>

This entry should be *split* into つく reading and はく reading.
I’ve added [restr] tags, but the entries would be much cleaner (and make it much easier to show further nuanced definitions) if つく／“take a breath” and はく／“vomit” were separated.

@@ -17,0 +17,1 @@
+<stagr>つく</stagr>
@@ -20,0 +21,1 @@
+<gloss>to take a breath</gloss>
@@ -22,0 +24,1 @@
+<stagr>つく</stagr>
@@ -27,0 +30,1 @@
+<stagr>はく</stagr>