Is my car engine damaged?

About 3 months ago I took my car in for a recall and while I was there I had also asked them to replace the air filter. As I had about a year that I had replaced the air filter. It's a 2013 ford fusion. They did the job, I signed the paperwork, paid for the filter and left. A week later I had started noticing that the engine sounded different when accelerating or starting from cold. I noticed that the intake that goes to the throttle body was not installed properly! It was loose and a whole bunch of unfiltered air was coming into the engine. I fixed it, but now my engine has been making some ticking and clanking noises when you accelerate a little bit. It runs fine and everything, but the dang "professionals" at the ford dealership didn't do their job properly. Could I have damage in the engine now?

Could it be damaged? Yes. Could you prove it was caused by the repair shop? No.

May be due to the recall. Take it back.

Ticking and clanking noises aren't form an air intake not being clamped tight.
It's likely because you haven't been changing the oil and filter every 3,000 miles or every 5,000 miles if running synthetic oil.
Most people don't even bother to check their oil level on the dipstick every 1,000 miles between oil changes.
When the oil level gets lower then halfway between the upper and lower dipstick marks you will have damage start in the motor.Motors with adjustable cam timing can have the cam actuators start binding up and throwing codes. Bearings in the motor wear.
A loose throttle body air intake tube would only allow unmetered air into the motor and make it run slightly leaner.

Probably not but I wouldn't be happy at all to find that after paying for the repair.

Not very likely, after all it is a FORD which stands for FOUND ON ROAD DIED!

You need a crystal ball. Unless you live in a dust bowl, one week with an air filter leak isn't going to wreck the engine. The noise may be some other loose things from sloppy work on the car.

How much dust and dirt is in the air where you live, if it was that bad you would have grown gills like a fish to filter the air.