"I want to read a book when it rains."
Translation:Yağmur yağdığı zaman kitap okumak istiyorum.
is there any way to express this using yağ(?)-nce instead of yağ-dığında?
Not really here. Using the suffix -(y)IncA implies that the action ends and then the second one starts. It would be like saying "I want to read a book once it rains."
I would oppose that. I think -(y)Inca is fine (exactly the same as -dIğIndA except that it's subjectless) and it doesn't necessarily mean that the action ends. Once actually would translate to these three forms.
Yağmur yağdı mı kitap okumak istiyorum.
Yağmur yağar yağmaz kitap okumak istiyorum.
Yağmur yağdığı anda kitap okumak istiyorum
And these forms in Turkish and once in English don't suggest that the first action has finished before the second action but it suggests that the second action started immediately after the first action.