(If this should be placed in a different forum, please let me know)

I am using Eudora on my MacBookPro. I put my address as one of my BB e-mail accounts. Pocket Mac doesn't support Eudora, so no synching.

Is there a way to tell the BB not to check e-mail on my Eudora account except when I request it. Now it is doing it automatically, and I have to delete messages manually that I've already read on my laptop.

Eudora is just a mail client... completely independent of your BlackBerry.
They don't ever talk to each other.

Your BlackBerry is getting mail from whomever your mail provider is, and it will get mail automatically. You don't control when the BlackBerry gets a mail delivery from your mail server. That's just the way BlackBerry devices are designed to function.