It was always my understanding that Hittite borrowed the cuneiform script from the Sumerians via Akkadian. This would prevent Hittite from borrowing lexemes from Sumerian unless Akkadian borrowed them as well. Now, in Akkadian, logograms are understood to be Sumerian writings that are pronounced in Akkadian (e.g., DUMU.MEŠ would be pronounced mārū/mārī; "sons"). I don't know Hittite, but this is consistent with the brief description of Hittite cuneiform on Wikipedia.
In round one of the 2016 edition of NACLO (North American Computational Linguistics Olympiad), Hittite is used in exercise F (PDF, start on p. 15). They give a phonetic rendering of a Hittite cuneiform passage, including:
- dumumeššu (line 5; "his sons"; written [DUMU.MEŠ-ŠU])
- dingirmešša (last line; "and the gods"; written DINGIR.MEŠ-ša)
(I added the writing from the edition of the larger context given by Beckman in JANES 14 (1982), 11–25. The words are found in lines Bi9' and Bi17' of that edition [§10 and 12].)
Now these words are clearly from the Sumerian DUMU.MEŠ "sons" and DINGIR.MEŠ "gods". But how is that possible, if in Akkadian these sign sequences are pronounced mārū/ī and ilū/ī, respectively, and Hittite did not have direct contact with Sumerian? Where is the fault in my reasoning? Or is the "phonetic rendering" given in the NACLO exercise incorrect?