After adding a support email address to your account, you would have to activate it, to receive tickets in freshservice from that email address. Kindly follow the below steps to activate your support email address.
1. Goto Admin-->Support Channels-->Email-->Add New Service Desk Email and add your new service desk email.
2. After the above, click on the send activation option under the recently added e-mail.
3. Goto your mailbox, click on the link within the activation email you would receive, to finish activation.