Is there any issue with hanging a working clock but it’s a bit broken in the back, or is it just a superstition? 

There is no Halachic issue whatsoever with doing so, nor is such a Kepeida or custom to be particularly cautious about this matter recorded in any of the Seforim which list a range of “superstitions” in the realm of Shemiras Hanefesh.

While there are Torah sources which mention to be cautious in matters that people consider worthy of caution, it applies to matters which are rooted in ancient Jewish tradition, but not to “beliefs” which are sourced in general folklore and have no Jewish basis.

See also the references listed here in the מראי מקומות tab.