Matsya Purana Chapter 53.:
The Vişnu Purana. Similarly, the rules of virtue laid down by Parāśara, for the period when there was the Varaha incarnation of Vişņu, in the beginning of Varaha Kalpa, form the Vişnu Purana.
That Purana contains 23,000 slokas and one who gives it away in charity, along with a cow and clarified butter, in the month of Aşädhas, on the full-moon day, is purified of all sins and goes and rests in peace in the region of Varuņa.
The Vayu Purana. The rules of virtue laid down by Vayu, along with the glory of Rudra, in the Śveta-kalpa, is known as Vāyavīya Purāņa. It contains 24,000 slokas.
One who gives it away in charity, along with a cow, jaggery and an ox, to a Brāhmaņa, whose family is large, on the 15th day of light half of the month of Śrăvaņa, attains the region of Śiva and resides there for a Kalpa.
But,Just because a purana belongs to a particular kalpa doesn't mean that in that Kalpa one should just read that puran only. There's no scriptural base for this.
All the Puranas depends on Mahabharata and one can get all benifits of 18 puranas. Basically, Mahabharata is the summary of all the 18 puranas.
MBH.:
1:01:67-71.: “Then the greatly glorious Vyasa, addressing Brahma Parameshthi, said, “O divine Brahma, by me a poem hath been composed which is greatly respected. The mystery of the Veda, and what other subjects have been explained by me; the various rituals of the Upanishads with the Angas; the compilation of the Puranas and history formed by me and named after the three divisions of time, past, present, and future; the determination of the nature of decay, fear, disease, existence, and non-existence, a description of creeds and of the various modes of life; rule for the four castes, and the import of all the Puranas”.
1:02:380.: “O ye Brahmana, as the four kinds of creatures (viviparous, oviparous, born of hot moisture and vegetables) are dependent on space for their existence, so the Puranas depend upon this history (Mahabharata)”.
18:06:97.: “One devoted to Vishnu acquires (through reading this Mahabharata) that merit which is acquired by listening to the eighteen Puranas. There is no doubt in this”.
I hope this clarifies all your queries. Prd..