I don't think there is a requirement for an additional DX code for the discontinued procedure. However, if you wanted to include a code from chapter 21: factors influencing health status and contact with health services Z00-Z99, you might consider the following codes:
Z53.09-Procedure and treatment not carried out because of other contraindication.
Z53.8-Procedure and treatment not carried out for other reasons.
I think based on the reason for the provider discontinuing the procedure stated in your original post that Z53.8 is probably most appropriate. I don't know if you can definitively say the procedure was contraindicated which Z53.09 indicates, I think you would need the physician to actually state it was contraindicated for whatever reason in the report and your provider only states that the procedure was not carried out because "tenodesis screw pushed right through the severe tendon fraying and would not hold".
Whether the physician's statement meets the criteria for the procedure being contraindicated, I'm wouldn't be comfortable making that judgement. What I do know is that the procedure was not carried out because it would not be successful since the provider stated the tenodesis screw wouldn't hold based on the severe tendon fraying. So, if I were in your position and felt like I had to use an additional DX to indicate why the procedure was discontinued, I would go with Z53.8.
Since there isn't anything that requires the additional code to explain why the procedure was discontinued, I personally wouldn't bother with adding a Z code to the claim and linking it to the line for 29828-53.
I happen to work for a commercial insurance company and our provider administrative policy for using modifier 53 doesn't indicate that any additional DX is required to indicate what the procedure was discontinued. The policy only requires that the clinical information documented in the patient's record must support use of the modifier and include a statement indicating
when the provider discontinued the procedure and
the extenuating circumstances preventing the completion of the procedure.