Yes, you can promote hits using the API. (In Algolia a 'hit' is where you can use Rules to raise some results over others, or put one result or banner at the top of the results.)
Here is how you can add rules from the API client: https://www.algolia.com/doc/api-reference/api-methods/save-rule/#save-a-rule.
Here is how you can promote hits to a certain position with the API client: https://www.algolia.com/doc/api-reference/api-methods/save-rule/#method-param-promote.
Additionally, here’s some more information about promoting hits using rules: https://www.algolia.com/doc/guides/managing-results/rules/merchandising-and-promoting/how-to/promote-hits/.