You can manage (i.e. create, update, delete) Test Executions directly through the JIRA's native REST API.
Xray REST API provides the following endpoints.
Test Execution Tests
To view the Tests associated with a Test Execution you need to specify the key of the Test Execution you want to view. To associate or remove a Test from a Test Execution you need to send a json with a list of the keys, when associating this keys can be Test keys or Test Set keys but when removing they have to be Test keys. To remove a Test from a Test Execution you need to specify the key of the Test you wish to remove.