I’ll break it into two parts.

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1. Can an abhavi / abhavya jīv do bhāv-prāyaścitta?

First, meanings in simple words:

• Abhavya jīv (अभाव्य जीव) = a soul which will never get samyak darshan, and therefore will never attain moksha.
• Bhāv-prāyaścitta = inner repentance: real feeling of “I have done wrong”, deep regret, humility, and a firm inner resolve not to repeat, done for śuddhi of the soul, not just for worldly benefit.

About abhavya:

• Abhavya jīv can do:

- vrats (vows), - tapas (fasts, penances), - dān, - external and even emotional “sorry”.

• But, because that soul is incapable of ever attaining samyak darshan, its inner “bhāv” never becomes truly samyak (right faith–oriented).

So:

• From vyavahār (worldly) viewpoint:

An abhavya can do something that looks like prāyaścitta: feeling sorry, asking forgiveness, doing fast, etc. It will give puṇya, better future birth, etc.

• From niścaya (real spiritual) viewpoint:

It is not the pure bhāv-prāyaścitta that leads towards moksha, because: - real bhāv-prāyaścitta ultimately rests on right faith (samyak darshan), - and an abhavya will never get samyak darshan.

So, in short:

> An abhavya jīv can perform the actions and even emotions called “prāyaścitta” in common language, but it never becomes that true, moksha-upāy bhāv-prāyaścitta which purifies darshan-mohaniya and leads to liberation.

You can read more on bhavya–abhavya difference here

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2. Can a mithyādṛṣṭi bhavya jīv do bhāv-prāyaścitta?

Here:

• Bhavya jīv = has capacity for moksha.
• Mithyādṛṣṭi = currently in wrong belief, no samyak darshan yet, but potential is present.

Such a soul:

• Can definitely:

- feel real guilt for violence, lies, greed, etc. - bow down, ask kṣamā, do pratikraman, - make a deep, honest internal resolve to change.

If the soul is bhavya, then:

• This kind of sincere inner repentance can:

- weaken wrong-belief (mithyātva), - become a powerful cause for future samyak darshan, - and gradually lead to higher guṇasthāns and finally moksha.

So, for your second question:

> A mithyādṛṣṭi bhavya jīv can do real bhāv-prāyaścitta, and it is very meaningful spiritually. > If done sincerely and repeatedly, under right Jain guidance, it becomes a strong step towards samyak darshan and moksha.

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Summary

• Abhavya jīv – may do prāyaścitta-like acts, but they never turn into that true bhāv-prāyaścitta which leads to moksha.
• Mithyādṛṣṭi bhavya jīv – can do bhāv-prāyaścitta in a real sense; this is a very important cause for spiritual progress and eventual liberation.