Just make sure to include something like, "I will be available to start working in May." This normally isn't a problem, since recruiters know some people already have jobs that they will have to quit. Some jobs might be more time-sensitive though, depending on how competitive it is and the nature of the job. Writing cover letters is stressful, isn't it!

What you wrote is fine. It is quite normal to have a period to work after resignation, so a delay before you take up an appointment is normal. . "I'm interested in the advertised position. My current employment contract finishes at the end of April I trust a start immdiately after that would be acceptable. or My current employment obligations are till the end of April. If appointed I'd be available immediately after that. . If appointed I'd be available immediately after my current employment obligations are completed at the end of April. .

I've seen both "cover letter" and "covering letter" being used, but I happen to see "cover letter" more often.

I've always called it a covering letter. Has the world changed?