Can I put a rack on a carbon road bike?

I suggest looking at frame bags. They don't carry much, but they have more capacity than the more sensible under-seat bags. You can get various sizes (up to 'fill the triangle'), and could even do both a frame bag and a seat bag.

There's a cycle tourist with a carbon bike a bit heavier than yours here so it can be done. Most of the carbon bike people tape their frames where the bag will rub to prevent the bag slowly grinding through the frame, but that seems to be the main concern.

I have a rack on Carbon Fiber Trek 5000 series frame. It's been there for around 7 years. I did it by drilling and tapping holes into the rear dropouts, which are aluminum. So none of the weight is on the frame at all — it's taken by the wheel directly. A single stay at the top anchors the rack to the brake bolt.

Now, I would not think of using the bike for a loaded tour. But for holding my briefcase on for a trip to the office: no problem at all.

Of course, many newer carbon frames don't have a flat place on the dropout where you could do this. A better alternative might be a racks that mounts to the axle, attached by the QR skewer.

Yes, you can put a rack on a CF frame, given the right rack. Namely, Tailfin makes racks explicitly designed to work with CF frames without rack eyelets, like "racing" road bikes.

You could get a beam rack that attaches to a seat post. Only issue is that you'll be limited by weight you can carry.

If you have a https://www.sjscycles.co.uk/carriers-racks/tubus-adapter-set-for-quick-release-axle-mounting/

https://www.tredz.co.uk/.M-Part-Seat-Clamp-Mount_34111.htm?sku=79197&utm_source=google&utm_medium=cpc&utm_campaign=google_shopping&gclid=Cj0KCQjwyJn5BRDrARIsADZ9ykEEpZRlvyZUEBcTRVOSNHR2I2tft31vzRVSBLG6GW9dD0YVcvufXaQaAlmbEALw_wcB

Then stress is transferred off the brittle frame. Toured like this on a carbon frame. Just be prepared for fun and games when changing an inner tube.