Compiling Hours for an Activity via the API

Posted on behalf of Colin Pickup. Original posting date 2015-09-10.

I am trying to compile the hours of support work done in the company workspace using the API. For this to work, I need to be able to link a timesheet with an activity or assignment using the API, but I can't see how these two might be linked.

Support is one of the activity types in our workspace, I need to sum the hours every month for each client. That is to say, this month 10 hours were spent on support for client X and 5 for client 5.

From the looks of things, I can see when hours were actioned using the timesheets, but these are not broken down by activity.
I can see the activity on the task assignment, and the number of hours logged, but it has no record of when the hours are logged.

Is there a way to link these two using the API?