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Hypothetically, secret data could be hashed in some schemes in one way or another. Could chunking of the data lead to exposure of secret component through side channel?

Say converting a randomized signature scheme deterministic by hashing the message with the private key to yield one-time variable (e.g. $k$ in ECDSA); or a post-quantum public-key encryption algorithm achieves semantic security using Fujisaki-Okamoto (藤崎-冈本) transformation, the ciphertext is bound to be related to hash output of the secret seed to be encrypted. In such situation, could secret be leaked?

Is there other way secrets can leak through hashing?

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