Pradyumna is the member of Sri Krishna's eternal quadruple expansions (catur-vyuha) and Kamadeva who is a devata entered into his body to acquire a body.
Jiva Goswami says that the statement that Kama who is burnt by Shiva took birth as Pradyumna is only partially acceptable and is a limited proposal.
For Pradyumna to be burnt or for Kamadeva to become Pradyumna is an impossibility since Pradyumna is famed as a member of Sri Krishna's eternal quadruple explansions(catur-vyuha) as sated in Gopala Tapani Upanishad.
yatrāsau saṁsthitaḥ kṛṣṇas tribhiḥ śaktyā samāhitaḥ |
rāmāniruddha-pradyumnai rukmiṇyā sahito vibhuḥ ||
There the powerful Lord Kṛṣṇa resides, situated with excellent
pastimes, accompanied by Balarāma, Aniruddha and Pradyumna and by his
śakti Rukmiṇī. Gopāla-tāpanī Upaniṣad 2.36
It is well known that Kamadeva is an ordinary devata and is constitutionally unfit to be included within quadruple expansion.
Therefore with the intention that will be spoken of later[SB 10.55.12] Sri Suka says:
kāmas tu vāsudevāṁśo dagdhaḥ prāg rudra-manyunā dehopapattaye bhūyas
tam eva pratyapadyata
The real Kāma-deva is an expansion of Vāsudeva. Cupid had previously
been burned to ashes by Rudra’s anger. Now, to obtain a new body, he
again merged into the body of Pradyumna. SB 10.55.1
avedajñasyāpi brāhmaṇye saty eva brāhmaṇas tu vedajña itivat tu śabdo'tra
"“Though a brāhmaṇa ignorant of the Vedas may exist, a real brāhmaṇa (tu)
knows the Vedas.” The word tu differentiates a true brahmana from one how is brahmana who is merely by birth.
The real Kāmadeva is an expansion of Vāsudeva. Or the word tu indicates a new topic. Stating
Vāsudeva’s aṁśa is Kāmadeva has the same meaning. He who previously was
burned by Śiva’s anger, the material devatā Cupid, in order to take a body
entered (pratyapadyata) Pradyumna, known as Cupid since it was impossible to
otherwise attain a body by himself because of Śiva’s curse. Bhūyaḥ indicates that previously he had arisen from Pradyumna.
Alternatively [the word dagdha can be red as adagdha due to sandhi], he[Pradyumna]
who previous was not burned (adagdhaḥ) by Śiva’s anger, again entered
Vāsudeva in order to reveal his form, in order to take a body for manifesting
pastimes in this world. He was not burned since he was Vāsudeva’s aṁśa.
What was said is made clear:
sa eva jāto vaidarbhyāṁ kṛṣṇa-vīrya-samudbhavaḥ pradyumna iti
vikhyātaḥ sarvato ’navamaḥ pituḥ
He took birth in the womb of Vaidarbhī from the seed of Kṛṣṇa and
received the name Pradyumna. In no respect was he inferior to his
father. SB 10.55.2
He who arose from Kṛṣṇa’s seed and who was named Pradyumna, he appeared,
born from Vaidarbhī, at the time of Kṛṣṇa’s pastimes on earth. This person was
not the other, material Cupid. The reason is given. He was not inferior
(anavamaḥ), he was equal to Kṛṣṇa his father in unlimited qualities and in form
(sarvataḥ). If it were otherwise he could not be said to be equal to Kṛṣṇa.
Though Arjuna is famous as being Nara in Mahābhārata, in the story of five
Indras, Indra becomes Arjuna. This actually means that Indra entered Arjuna.
Consequently Narada's instruction to Rati[10.55.6] to accept Pradyumna as her husband and her attaining him was not inappropriate. Even if we accept alternate meaning of the previous verse['adaghda' reading], by the strength of Nārada’s instruction, it is understood that the burned up material Cupid had entered Pradyumna. Rati’s qualification for directly contacting Pradyumna is like being touched by a touchstone.