As Aram states, turning on additional BESClient debugging is not likely to help the situation, as it’s a transient problem likely related to your client or the network, and not to Bigfix itself.
You’ll need to be looking for other clues in the Event Logs of your server & relays, looking at your firewall/router logs, network switch logs, etc.
If left alone, does the client ever resolve itself without your intervention?
From your other post I saw there are ten network hops from your client to your relay; that is likely a case where I’d recommend locating a relay closer to your client, or enabling Persistent Connections from your client to the relay, depending on your network architecture.
What is your total deployment size, total number of relays, number of locations, number of endpoints per location?
You mentioned earlier you need better guidance from support…to be clear, this is not a support forum but a peer enthusiast / self-help forum. I don"t know whether anyone from the Support team reads the forum, and if so it’s just on their own time - responding on the forum is not anyone’s job. For actual support, you’ll need to work with them through the PMR / Support Requests process. I’m honestly not sure how much that’ll help, diagnosing what’s going on with your network might be beyond their scope; the winsock errors could possibly be useful from a developer standpoint, but often a Winsock error just means “I could not connect” which is less-than-helpful. The codewords that go with the number are often more useful than the number itself - SOCKET RECEIVE, or BAD SERVERNAME, etc.
BAD SERVERNAME usually means your DNS isn’t working properly. SOCKET RECEIVE usually means your network or operating system is dropping packets along the way. The first doesn’t usually fix itself, unless maybe you are seeing that while rebooting DNS servers; the second usually resolves itself with retries in time.