have a look at the patch api under the heading of Execution
here is an example of a curl command I picked up from a colleague
curl -X PATCH --user ******@ca.com:ppppp --header "Content-Type: application/json" --header "Accept: application/json" -d "{ \"allowedStatuses\": [ \"RUNNING\" ], \"denyStartReason\": \"PHASE_CONTAINS_TASK_WITH_MISSING_PARAMS\", \"endDate\": 0, \"id\": 7, \"percentCompleted\": 0, \"phaseId\": 34, \"startDate\": 0, \"status\": \"RUNNING\"}" "http://$CDD_IP:8080/cdd/execution/0000/v1/releases-execution/7/phases-execution/34"