Yes. Current Moodiy hosting plans support custom domains, subject to the plan and domain workflow available in the client portal.
When a site is first created, you may use a Moodiy-managed subdomain. To use your own domain, submit a custom domain request from the client portal and follow the DNS instructions provided for that request.
A custom domain is not active immediately. DNS needs to be configured correctly, then Moodiy needs to validate the request and complete the required routing and SSL steps. Until that is complete, keep using the existing site URL.
If your domain is already connected elsewhere, or you are changing from an old domain to a new one, contact support before changing DNS so the transition can be planned.