We recently offloaded few users from office 365 to on-premises exchange 2010 and 2013 in environment however outlook clients are randomly being facing sync issues and unable to update back from on-premises . Here are the logs we found in sync issue folder. Remark issue comes only when users users are offloaded from o365 however when i create a new mailbox for these users and import back pst this is not the case. Please suggest. 

11:25:47 Error synchronizing folder

11:25:47                                 [8004010F-501-8004010F-560]

11:25:47                                 The client operation failed.

11:25:47                                 Microsoft Exchange Information Store

11:25:47                                 For more information on this failure, click the URL 

How did you prepare this offboarding?  Did you already have a hybrid configuration or did you create a new one and configure the directory synchronization?  Are the URLs pointed to on-premises Exchange?

Hi,

What is your outlook version ?

In your post, I understood you recreate a new mailbox and the issue won't occur.

Is the idea right?

According to the error code, I think the issue is related to the OAB.

Please check if oab file is blank on CAS server as following location:C:\Program Files\Microsoft\Exchange Server\ExchangeOAB\. If oab file is blank, please restart the MSExchangeFDS and check the result.

On your server side, please run the following command to check your OAB:

Get-OabVirtualDirectory | FL Identity,name,*URL*,*auth*

Here is these similar threads for your reference:

https://social.technet.microsoft.com/Forums/exchange/en-US/8cc08bf3-4a92-412f-8090-f3eae660136a/exchange-2010-outlook-2010-client-error-in-synchronization-log?forum=exchange2010

https://social.technet.microsoft.com/Forums/exchange/en-US/79b40a4b-ea8f-4fd9-b060-70dd8ba4b4ba/error-synchronizing-folder-8004010f5018004010f0-exchange-2010-and-outlook-2010-multiple?forum=exchange2010

Best Regards,

David