It doesn't matter what the client is, the fundamental question is "Can you move the mouse cursor on your PC from anywhere without installing software"
The short answer is no - which is a good thing! Otherwise, anyone on your network could just take control of your PC...
That said, you could, in theory, create an RDP (Remote Desktop) connection and use that to control the PC as a whole - but that is very complex, has been done already and would still require the user to allow remote desktop connections to the computer (Control Panel->System->Advanced->Remote)
Edit - Bare minimum app:
There are a number of ways to approach this but the absolute simplest app I can envision involves having an application on the PC listen for connections on a TCP/IP port. You'd then send messages to this port from Android using the Sockets classes.
The app would receive these messages, parse them and perform the appropriate mouse actions.
Make sure that you include an authentication/authorisation mechanism - you don't want random strangers to be able to control your PC just because they broke your wifi.
You may find it easier to build the desktop app to accept messages using the HTTP protocol (RFC) - This is a standard, widely used and very flexible mechanism for client-server communication. Why reinvent the wheel? This would also make your Android-side code far simpler as you could use HttpURLConnection and other similar classes which abstract the complexity of managing sockets.
You may also want to consider if the app should provide any feedback to the client - eg the new mouse position or a success/failure.
NB: Running the app as a windows service or website might seem preferable to a desktop app (doesn't need to be started by the user, nothing in taskbar/system tray) but there are considerable drawbacks to both - Windows services can't interract with the desktop easily (what happens if nobody is logged in?) and websites run as a different user so in addition to not having the same desktop, they have limited permissions.