I have a Powershell script that is used to remotely call other Powershell scripts on other servers. The script is used to shut down and start up services on the different serve
Sounds like a double hop problem. These are notoriously difficult to work around, since the credentials you would pass can't be authenticated by the second system.
CredSSP is a solution, but it does increase security risk so use caution, make sure you understand the configuration, and make sure you configure it right.