Have set up github in jenkins configuration, providing:
Let Jenkins auto-manage hook URLs
API URL: http://github.mydomain.com/api/v3/
username: jenkins
Password: <the jenkins user's git password>
OAuth token: <the jenkins user's oauth token>

Clicking Test Credential says 'Verified'

When I add a new job that connects to a repo on our github
http://github.mydomain.com/MYORG/myproject.git

and to only build:
Build when a change is pushed to GitHub

It doesn't get registered in the github enterprise hooks for that repository. Have ensured the jenkins user is in MYORG/owners team in github, which as full read/write/admin access.

If I manually add the jenkins hook to the webhooks in github enterprise, it successfully kicks off builds when I push to github.

Have tried turning on logging in jenkins to trace what is happening, but there doesn't seem to be any errors or warning anywhere in logs coming back:

Have added logger with ALL on:
com.cloudbees.jenkins.GitHubPushTrigger
com.cloudbees.jenkins.GitHubRepositoryName
org.kohsuke.github

but the only output I get is:
Jul 11, 2013 12:50:43 PM com.cloudbees.jenkins.GitHubRepositoryName create
FINE: Object is GitHubRepository[host=github.mydomain.com,username=MYORG,repository=myproject]
Jul 11, 2013 12:50:43 PM com.cloudbees.jenkins.GitHubRepositoryName create
FINE: URL matches java.util.regex.Matcher[pattern=https?://([^/])/([^/])/([^/]+)\.git region=0,47 lastmatch=http://github.mydomain.com/MYORG/myproject.git]
Jul 11, 2013 12:50:43 PM com.cloudbees.jenkins.GitHubRepositoryName create
FINE: Constructing from URL http://github.mydomain.com/MYORG/myproject.git
Jul 11, 2013 12:50:43 PM com.cloudbees.jenkins.GitHubRepositoryName create
FINE: Object is GitHubRepository[host=github.mydomain.com,username=MYORG,repository=myproject]
Jul 11, 2013 12:50:43 PM com.cloudbees.jenkins.GitHubRepositoryName create
FINE: URL matches java.util.regex.Matcher[pattern=https?://([^/])/([^/])/([^/]+)\.git region=0,47 lastmatch=http://github.mydomain.com/MYORG/myproject.git]
Jul 11, 2013 12:50:43 PM com.cloudbees.jenkins.GitHubRepositoryName create
FINE: Constructing from URL http://github.mydomain.com/MYORG/myproject.git