What is the ruling on dealing with a mother who has secretly embraced Qadianism, and is it permissible for the father to continue the marital relationship with her?

The Qadianism is a misguided and disbelieving sect. If you are certain that your mother follows this misguided belief, you must strive to bring her back to the true religion by presenting arguments. It is preferable to seek the help of specialized scholars in the study of sects and doctrines.

As for her relationship with her husband, scholars have agreed that the husband should not approach her once she apostatizes. They differed on whether the marriage remains until the end of the waiting period (iddah) or dissolves immediately. The more accepted view is that if she returns to Islam before the end of the iddah, the marriage remains valid; otherwise, it is nullified at the end of the iddah.

It is stated in Imam Al-Shafi'i's book, "Al-Umm": "If the husband apostatizes after consummation, he is separated from his wife. If her waiting period ends before the husband returns to Islam, the marriage is dissolved. If the wife apostatizes, or both apostatize, or one after the other, I always consider the waiting period. If it ends before they both become Muslims, I dissolve the marriage. But if they embrace Islam before the waiting period expires, then the marriage remains valid."

The wife during the waiting period is not entitled to maintenance (nafaqah). It is stated in "Al-Kafi": "If a Muslim woman apostatizes, she has no maintenance, similarly."

And it is stated in "Al-Hawiyy Al-Kabir" by Al-Mawardi: "If the wife apostatizes, and it was before consummation, she has no dowry. If it was after consummation, she is entitled to the full dowry, but she has no maintenance during the period of apostasy, by unanimous agreement... If she does not embrace Islam until the waiting period has passed, the marriage is nullified. If she embraces Islam before its expiry, they remain married, and she becomes entitled to future maintenance after embracing Islam."