If your eM Client V7, V8 or V9 is receiving and sending mail ok as you advised, then suggest to first go to “Menu / Accounts” on your HP Laptop that is working ok and click on your account on the left and “write down all the mail server settings on each tab” along the top. If you have more than one account, write down all of them so you can then put those in the V10 computers.
Then “if the problem V10 account is an IMAP, Exchange, Office 365 or iCloud (Non POP) account”, then just remove and re-add your email account using either the “Automatic account wizard at the top” or add manually via the “Add Account / Mail” options depending on what your mailbox provider supports. Then once you have finished adding the account, before saving & exiting accounts, “check that you have the correct settings as you wrote down” in all the tabs along the top in your Account. All your emails will then update automatically from your mailbox server end when eM Client syncs. You won’t loose anything as its all server based. The only thing you will normally notice is that the emails will be slower to read till fully cached.
Now “if you have a POP account”, then first create a folder down in eM Client “Local Folders” at the bottom left called eg: Old Mail, and then “Drag / Move” all your current POP account mail and any custom POP account mail folders etc down under Old Mail. Then once you are certain they are all there, then remove your old POP account. Next re-add your POP account via “Add Account / Mail / Other” option and use the settings you wrote down. Last thing once you have completed setting up the account again as mentioned above, check the settings are correct “via the tabs along the top”. Then save and close accounts and see if that fixes the problem.
Note:- Before doing anything, make a manual backup of eM Client via “Menu / Backup” incase you need to restore for any reason. You can see when the backup is complete via clicking the dropdown on the right of Refresh top left and clicking Show Operations.
Lastly if removing and re-adding your account “with the same HP Laptop V7, V8 or V9 settings makes no difference”, then suggest to uninstall eM Client and if you have Windows, “don’t delete the database when asked on the uninstall wizard”. Then download the latest eM Client V10 from the release history page and reinstall and see if that fixes the problem.