To send from your gmail account while keeping your email account displayed when sending like : firstname@yourdomain.com, you must do this from the gmail console: settings/accounts and import/'Send mail as'.
After, you must configure cPanel to use your email accounts with Gmail service, see our autoconfig plugin : Remote MX Wizard, in your cPanel sections Emails/Remote MX wizard. Configure the desired domain.

Manual method and Gsuite /ex : refer to the provider's documentation in the first place.
From your cPanel, in domain section, see Zone Editor, you can here modify SPF and MX entries.
(to make clean changes, the recommended method is by cPanel/Domain/Zone editor, OR mail section/'Email Deliverability' with custom SPF, you can add IP, MX, include, etc... and click 'update')
SPF :
Depending on whether you are using an external provider for your mails, you may need to modify your default SPF (TXT v=spf1...)

MX :
The priority works with 0, if you want to switch to external, add any new entry with priority 0, the entry you had by default must now have minimum priority 10-20-30 etc ....

Email routing, in section Email, you should normally put for the concerned domain : 'Remote mail server'

To test your SPF and DKIM, use this testing tool :
https://www.port25.com/authentication-checker/ Send a test mail from the email account you want to test:

If your mail is: mymail@mail.com

You must send the mail test this way : check-auth-mymail=mail.com@verifier.port25.com 

You will receive the report on this email, which counts 'pass' for the SPF and DKIM, ignore the rest...

Old Prestashop versions issues with PHP and Gmail

It is because before version Prestashop version 1.6.1.5, it uses an old version of SwiftMailer. This causes Socket Error with gmail smtp.

The fix - upgrade prestashop OR upgrade swiftmailer (with this free module):
https://www.prestashop.com/forums/topic/519767-free-module-swiftmailer-upgrader-version-541/ 

(This fix worked by example for prestashop 1.6.0.8 with PHP 5.6)