When you say the process has been Paused or Stopped, has that been done in Workspace?

If so, you can either use Workspace itself to Start the process again (not Restart as in Start from scratch, basically Continue from the last position) or you can use the SourceCode.Workflow.Management API. If you use the  API, you will need to have the Process Instance ID before hand though.

Have a look at the Developer Reference that gets installed with the K2 Help File, it will have theObjects required documented. In particular, the StartProcessInstances method of the WorkflowManagementServer object.