Lug nuts can't get the hub cap off, or the scrws tht hold the hub on

Unfortunately, removing the hub will damage the hub bearing and ABS tone ring as they are pressed together. There are aftermarket studs that are able to slip into the empty stud hole but you have to modify the back of the stud to clear the tone ring. Using a brass drift and a dead blow mallet, strike the end of the stud to push it back in till it clears the splined area. You may have to shorten the broken stud with a cutting tool to get it out past the tone ring. Insert the new stud into the hub and using a half inch washer and lug nut, tighten the lug nut with an impact wrench till it seats on the back side of the hub. The lug nut must be a standard lug nut--- no threaded cap style or chrome style will work as the lug nut is installed BACKWARDS to avoid damaging the chamfered end.

If they are hubs, they should just pop off.

The plastic tool you have removes the hub cap. There is a metal end on the one side of the tool to remove the center cap. Pry it off. Then the tool will fit in to the plastic nut that holds the hub cap on.

Most hub caps close to that year Honda are held on with the lug nuts, you will have to remove the wheel lug nut's to remove hub cap.

#1 Look carefully at the CENTER of the hub cap. Gently pry the center cap off. The rest will be self explanatory.
Or
#2 Take one lug nut off . Sometimes there is an extra set of small lugs to hold the hub cap on.
I do apologize for the two choices. Just cannot remember which way the Cougar works.

Removing: Loosen lug nuts (7/8") and spindle nut (36mm). Lift truck. Remove lug nuts, spindle nut, and wheel. Remove 2 bolts (3/8" allen) holding calipers on. Hold the calipers up with a bungee cord. I attached it to the sheet metal of the fender. Remove 4 bolts (15mm) from rear of hubs. Once these bolts are removed you can move the shield that is in-between the hub and the axle(I think). You are moving it to get access to the allen bolt that holds the ABS sensor on. Once you move the shield then you can look through the top of the rotor through the slots and see the sensor line and bolt. I don't remember what size allen this one took. You do need a long allen wrench though. Disconnect sensing line from hub. Now the rotor is ready to come off. I had to use a pry bar to remove this gently working each side until it popped off. Once the rotor is removed, you need to hammer out the lug bolts. Then you have to tap the rotor away from the hub. This took a little work. Then check each of the lug bolts by screwing the lug nuts on them. They should go all the way down by hand. If they don't - go buy new bolts $2 a piece. Installing: Position new rotor on hub. Get some dry ice from your local grocery store and freeze the bolts for about 10 minutes. Pull one bolt out of the ice at a time and install on the rotor and tighten the bolt with the lug nut. I put the nuts on upside down and used a 7/8" wrench and mallet to tap on the wrench to get these as tight as possible. You also have to tap on the head of the bolt and repeat the wrench tapping. Do this for each bolt pulling one out of the ice at a time. You do not have to get the rotor and hub completely together but get it as close together as possible. Once all of the lug bolts are in it is time to put the rotor on the truck. Ensure the shield is on the spindle and install the rotor, again, this takes some work. Once the rotor is on, connect the ABS sensing line. Line up the shield and install 4 bolts to rear of hub. You can align these by moving the hub, not the rotor. Install the caliper, remember to put brake grease on these 2 bolts. You may have to compress the caliper slightly. To do this remove the brake pad that has the circular part of the caliper touching it. Open the brake fill cap on the master cylinder. Use a C-clamp to compress the circular part. Reinstall the brake pad and brake fill cap. Install the caliper. Install the wheel and tighten lug nuts. I used a cheater bar on my tire iron and ensured that the lug bolts were flush with the head of the lug nuts. Then I removed the wheel to verify that the gap between the rotor and the hub was gone. Reinstall wheel, tighten lug nuts, and tighten spindle nut. Once both brakes have been replaced, start up the truck and pump the brakes several times till you feel some tension when pressing them.

It may just be a stone that got inside the wheel and hasnt gotten tossed out. It could also be a loose lug nut. A good visual inspection should be able to find the problem. One thing to note, the black plastic caps that hold your hub cap on are not your lug nuts! They screw onto your lug nuts, so to see the lug nuts you'll need to unscrew these caps.

the lugs should have a flat washer and a spring washer the lug should have a long cylinder shape on one end that is tapered to tighten up the wheel the washers hold the plastic hubs in place you might try a wrecking yard or a dealer for them

there is a large nut that holds that hub on. Remove the cotter pin and the cap from the bolt... remove the bolt and the hub comes off.

Robert