Kind of. You have it mostly right, at least, if I read you right. I’ll elaborate and seek to clarify to be safe.
What you can claim depends on what would be considered novel and non-obvious relative to the original patent, and any other prior art – it doesn’t matter whether or not a patent exists and others’ claims are actually besides the point. The concept of attaching a thatching rake to a lawnmower, if novel, might or might not be considered obvious. But even if so, there is probably something you could change to obtain a patent, i.e. there’s probably a decent chance that you could not obtain a patent that would prevent someone else from marketing any random lawn mower / thatching rake combo, but you could more than likely still obtain a patent.
What you can manufacture/market without infringing depends on the original patent’s claims, and not at all on whether or not you can or will obtain a patent. I should clarify that if the original patent has not expired, and you are doing nothing more than attaching a thatching rake to a completely identical lawn mower, you are most likely infringing once you do it (although it’s possible you are not, due to how the original claims are written). But depending on the claims, hypothetically it’s possible that you can avoid infringing the original patent by, say, removing the blade and replacing it with a different blade. Or not. That might be besides the point, because as Julie mentions, you can sell your attachment separately. In that case you should be conscious of any trademark related concerns, and it’s hypothetically possible that the manufacturer of the lawnmower could give you some legal headaches even if there is no legitimate grounds for them. I know that Apple is known for that.
The other possibility is that you could license the original patent from the patent holder. It is worth noting that if you buy a lawn mower @ Hope Depot per se, generally it is considered that you have a license to do what you want with that lawn mower. But if you tried to add your attachment and re-sell, you would have to very conscious of potential trademark issues which would likely arise if people could confuse your modified mower as being manufactured by or sold by or otherwise endorsed by the manufacturer/brand of the original mower. That could actually still be an issue even if you were to manufacturer your own similar lawn mower.
