Could you be a bit more descriptive in what you are doing. What do you mean by "Can photos be exported to CC from Lightroom Desktop?" What is the CC you are referring to.
I doubt a Spark page in a web browser has the ability to access Lightroom that is installed on your computer.
In LR Export the image/s you want to use then point Spark to them where you saved them on your computers hard drive.
If you have File Sync turned in your creative cloud app (the gear icon->preferences->Creative Cloud->Files), you can export images from Lightroom to your cloud folder structure by simply exporting from Lightroom into the cloud synced folder and your computer will automatically upload them. Strangely enough there is no native publish service or plugin to do this in Lightroom but the export works just fine. I haven't used Spark but I would assume you can then see them directly there.
Thanks for the response. The mobile lr does sync directly to the cloud. The desktop version should follow suit.
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Can you be a bit more verbose about
- where and where not you think LR Mobile is syncing
- where and where not you thing LR desktop is syncing