Are you allowed to read prayers (to catch up) while kaddish is being said? How about in between amens, yehey shemey raba and the ending amens?
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The Mishnah Berurah (56:1) says that it is forbidden even to think about words of Torah during kaddish. We can make a kal vachomer that it is certainly forbidden to recite words of Torah or prayer during kaddish.
It also gives a number of stories about what evil befalls people who talk then, mentioning "the middle of kaddish", "yisgadel", "when the chazzan is saying kaddish", and "responding to kaddish", implying that one must be especially careful regarding talking during the "yisgadel" part and when responding amen. However, the very strong language used suggests that talking during any part of kaddish is wrong.
Even if one is in the middle of Shemoneh Esrei, one should pause when kaddish is being said (Rashi to Sukkah 38b). Certainly, one should not coninue with other prayers during the recital of kaddish.