A “Delta HDF Faucet?”

First of all the model that you mentioned does not correspond to the picture that you have on there unless it is a much older model it does however fall under similar faucets that I have worked on myself. The cap you mentioned that you can't get off does have to come off. Right there where the grooves are there is a joint that unscrews that cap. Yours has just gotten so tight you can't get it by hand. A pair of channel lock pliers and a rag to try to prevent any marring if possible should do the trick. After you get that off there will be a little grey or white plastic cover with an odd shaped hole in it that that brass stem comes through. That cap comes off and then that stem is attached to a metal or plastic ball that has holes and or slots in it. It just rests in that . You must remove that plastic cap and then you will see two or three holes with rubber cups and springs in them. Remove the cups and the springs with needle nose pliers. You will have to take all the parts with you just removed to the hardware store and they will have Delta parts or similar replacement parts made by other companies that will fit your faucet. Just match them up to what you have removed from your faucet and take them back home and put them in reverse order and you should be good to go of course you have to turn off the water to that faucet which usually means you have to turn off the water to the house. Oh yes and by the way if you do have to mark up the original chrome cover with the grooves on it in your picture with the pliers getting it off a replacement usually comes in one of the kits that you will be buying. Good news is it is pretty simple once you get that super tight cap off.

Looks like all I needed to do was wait a little longer. The WD-40 did its job and there was lots of corrosion in the spigot at the bottom where the spigot ring met the spigot.