How to integrate TestRail with Jenkins?
There is a Jenkins plugin maintained by the Jenkins team at GitHub:
Integrate test results from Jenkins into TestRail. Upload your junit test results to TestRail after every run.
The plugin only allows to populate TestRail with the test results, but maybe you can start from there.
I too am attempting to integrate TestRail with our Jenkins installation. This blog post is pretty good: https://qxf2.com/blog/reporting-to-testrail-using-python/
You will need a way to map each Jenkins test to the appropriate TestRail test-case id, Ex: C123.
After the Jenkins run, reconcile the test results with the mapping and POST to the TestRail API with the test_run_id, test-case id, and the test result status_id as defined in TestRail.