Under GitHub Pull Request Builder configuration Auto-manage webhooks is enabled.

In the GitHub Pull Request Builder added user credentials. User belongs to owners' User's credential tested and are working.

Webhooks management however does not work. For every private repository there's an error message displayed in Jenkins: 

There is no credentials with admin access to manage hooks on GitHubRepositoryName[host=github.com,username=ORG_NAME,repository=REPO_NAME]

From the error message it looks like plugin is trying to use organisation name as a user name, so authentication fails.