Assuming you mean the traditional Lightroom Classic when you say Desktop, then......
If you want so sync TO Lightroom Mobile, then you will need to create Synced Collections (only lower res Smart Previews will be uploaded to the Cloud). Meanwhile, any photos imported into the Mobile Device (or taken with the in-app Camera) will sync full resoluion to the Cloud and, if sync is turned on in Classic, will automatically download these at full resolution to the folder desigmated in Classic's preferences.
If you mean Lightroom (Cloud Based) Desktop, then any photos imported via this, Mobile or Lightroom Web will sync full resolution to the Cloud, and be available across the whole Cloud Ecosystem at full resolution (although depending on many factors, the full res won't actually be downloaded on your device unless required.)
It isn't quite that straightforward. If you install Lightroom Classic you can migrate one catalog to the cloud. If you do this then the cloud version sort of "takes over" and is in charge. If you don't migrate the catalog then you can synchronize collections from a single catalog. This will allow you to create collections in Lightroom Classic, and choose to synchronize those collections. When they are synchronized this will send smart previews to Lightroom for the cloud. You will be able to edit those smart previews on any mobile device or other device that can connect to your Lightroom account, and those edits will synchronize back to Lightroom Classic. If you take photos using your smart phone those images will synchronize back to Lightroom Classic.
I have not migrated a catalog to Lightroom for the cloud. That is my choice. I prefer Lightroom Classic as my primary version. You might want to get other opinions and perspectives before you make a decision on what to do.
Thank you Jim, I agree with the your editing choice but had wondered if this was a reasonable expectation. I have lots of files that I would like to cull and a mobile option could be helpful, but not at a risk of redoing what was already done.