I've been asked this question and I've been unable to get an answer from my other sources, so perhaps some of this lists members might know.

I was asked whether based on this half verse
kālarātrirmahārātrirmoharātriśca dāruṇā  there are three main nights of worshipping the Goddess or four nights. (I've traced that half verse to the Markandeya Purana, and list the section containing that half verse below).

Specifically, I've been asked:
1) Based on this half verse are there 3 or 4 nights of the Goddess?  note: all the translations I've seen have dāruṇā in this half verse as an adjective, but I've been told that some people says its the name of a night of the Goddess, daruNAratrI ..
2.) Is mahAratrI  the night of mahAzivaratrIi, and Moharatri the night when Lord Krishna was born, and Kalaratri the night of Deepavali?
3)  Is there a fourth night of the Goddess, Darunaratri the night of Holi Purnima when the demon Holika was burnt?

I traced that half verse to the Markandeya Purana 81.59. In this section Brahma is praising the Goddess to awaken Vishnu from his sleep.  The section containing this half verse in the Markandeya Purana is:

brahmovāca
viśveśvarīṁ jagaddhātrīṁ sthiti-saṁhārakāriṇīm  /
nidrāṁ bhagavatīṁ viṣṇoratulāṁ tejasaḥ prabhuḥ  // MarkP_81.53 //
tvaṁ svāhā tvaṁ svadhā tvaṁ hi vaṣaṭkāraḥ svarātmikā  /
sudhā tvamakṣare nitye tridhā mātrātmikā sthitā  // MarkP_81.54 //
ardhamātrā sthitā nityā yānuccāryā viśeṣataḥ  /
tvameva sandhyā sāvitrī tvaṁ devi jananī parā  // MarkP_81.55 //
tvayaiva dhāryate sarvaṁ tvayaitatsṛjyate jagat  /
tvayaitatpālyate devi tvamatsyante ca sarvadā  // MarkP_81.56 //
visṛṣṭau sṛṣṭirūpā tvaṁ sthitirūpā ca pālane  /
tathā saṁhṛtirūpānte jagato 'sya jaganmaye  // MarkP_81.57 //
mahāvidyā mahāmāyā mahāmedhā mahāsmṛtiḥ  /
mahāmohā ca bhavatī mahādevī maheśvarī  // MarkP_81.58 //
prakṛtistvañca sarvasya guṇatrayavibhāvinī  /
kālarātrirmahārātrirmoharātriśca dāruṇā  // MarkP_81.59 //
tvaṁ śrīstvamīśvarī tvaṁ hrīstvaṁ buddhirbodhalakṣaṇā  /
lajjā puṣṭistathā tuṣṭistvaṁ śāntiḥ kṣāntireva ca  // MarkP_81.60 //
.. . . . 

Thanks,
Harry Spier