I think it would be prudent to create a new custom taxonomy now named podcast in order to categorize your episodes separate from other content.
Whether using a custom podcast taxonomy or generic categorys, taxonomies do not support attached media or featured images out of the box. You’ll likely need to store a media attachment ID for the image in taxonomy meta-data, and either expose that meta to the REST API using register_meta() or add it as a field to the relevant response using register_rest_field().
If podcast logos are some feature supported by the Podlove plugin/service, then you will need to consult their official support channels for how you should retrieve it via API.
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