Yours is an excellent question — actually a series of questions — that raise several different issues.
First, did your employer, the private care agency, have the right to reassign you to care for another of its clients rather than the one you expected to work for?
The answer to that question is likely “yes.” Absent an agreement to do otherwise (whether made with the individual employee or a union representing the employee) an employer generally has the ability — within reason — to direct the time, place and scope of the employee’s work.
Does the employer have the obligation to share a cancellation fee with you?
No, the employer does not, at least in the absence of an agreement with you or a union to do so.
Does the employer have the right to charge a cancellation fee to the SU when it is not paying its employee the some or all of the cancellation fee?
It’s hard to provide you with a definitive answer because we don’t know what its arrangements are with its client, or if its arrangement is with an insurance company or government agency.
An employer does incur some expense when there is a last minute cancellation, such as the cost of notifying its employee, and as in your situation would have incurred the obligation to pay you a cancellation fee had it not found another assignment.
Whether it can charge the SU or an insurer or government agency the full fee — or indeed any fee — in such a situation depends on its contractual arrangements with the SU, insurer, or governmental unit.