How should one tie his shel yad on his hand if his finger(s) have something on them, such as a splint for a broken bone in the left middle finger?
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The Rama rules (OC 27:4 based on Teshuvat HaRashba 1:827) that there is only a problem of a chatzitza (separation between the tefillin and the skin) under the boxes not the straps. The Mishna Berura there (sk 16) notes that most later authorities only accept this for the parts of the straps that do not directly relate to the tying (ie. the parts between the box and the knots). In your case, where the potential chatzitza is very far away from the box, it shouldn't be much to worry about.
Furthermore, Rav Ovadia Yosef rules (Yabia' Omer 2 OC 2) that one need not remove his watch when putting on his tefillin because it is far enough away from the box. Seemingly he would agree that in your case (especially since there is a medical need) you may be lenient and not worry about a chatzitza.
Shalom, If you look in the Mishna Brurah quoted above (OC 27:4 MB sk16) you will see that you can be lenient with regards to chatzitzah in the place of the "krichos" but whatever is shayach to the "keshira" you should be machmir .
The question you have to ask is what is the distinction between what is shayach to the keshira vs what is shayach to the krichos?
It was pointed out to me that is OC 25:11 Mishna Brurah sk38 he brings the Tshuvas Mishkenos Ya'akov and says "his position is also like our minhag to initially wrap on the arm for chizuk since without that it will not endure and is not called "keshira". According to this MB it seems like all of the wrapping you do on your arm is shayach to the keshira and therefore you should be machmir not to have a chatzitzah.
You could make a chiluk between the two Mishna Brurah's but at face value it requires some work.
Now I would have said like Double AA but because of this MB I would want to do some more research before I would confidently state the MB's psak.