Difficulty in turning a single handle faucet from side to side. Under the sink there is a barrel shaped weight that has come off and appears to have been wrapped around one of the hoses. Your assistance is appreciated.

Alot of variables there. If it's really old you must be extremely gentle. I have had them crumble into pieces in my hand. If it doesn't appear corroded around the edges and is in good shape, you should rock or pry gently with a non marring tool (putty knife covered by rag) alternating sides to break it loose. A penetrating oil like PB blaster can be used if you can let it sit for a day. Just straight pulling is not wise. Tapping (not pounding) can be helpful. You are just trying to break the bond between surfaces so the vibration of lots of light tapping will sometimes do it.

First, turn off the water supply under the sink. Next, put the faucet lever in the up position, no water should be coming out. While in the up position turn slowly while looking at the underside of the handle and you should be able to find a set screw. It will probable be an "allen head" type and you will need and allen head wrench to loosen it. An allen head looks like a hex shape, 6 sided I believe and is rod shaped. If you don't have one they come in sets at your local hardeware.

you will have two mounting nuts at each end and two water line to disconnect for removal under the sink on any faucet made

most times the retainer nut can be turned by hand or pliers and there is also a tool made to remove them
dont remeber the tools name but it has a crown on top and is tube shaped.

Hot water line connected to left side of faucet? Did you replace tap set, or just cartridge? Cartridge may be out 180 degrees.

check that nothing blocking hose return weight under sink

Remove spray head then faucet tighten connection on the spray hose

sounds to me like you did not engage handle properly the moen has like a half moon in handle that has to go on just right and can be a little tricky if it went on easy its not on right

Moen is a cartridge style unit. They make two different cartridges that you can buy at Home Depot. Turn the water off under the sink and then remove the handle. You will see the the cartridge. It has a U shaped retaining clip that pulls out. Once you remove it, then take a pair of pliers and pull the cartridge out. Take it up to HD and match the one you have. The new one will slide right back in and you will reverse the process.

Your problem is a simple one.

1. Turn off water to faucet.
2. Place a towel(s) in the base of the sink(s) to prevent any parts from going down drain.
   
3. Remove the handle to the faucet. (single handle only)
   
4. If not a single handle then remove spout from faucet.
5. If a single handle remove cap carefully and any loose parts.
6. Gently turn spout side to side while lifting to remove.
7. You will see a hole with a very small valve where the water comes out to the spout on the base of the faucet.

1. Remove this valve (which is a backflow preventer) and replace.
2. Reassemble and test.
   

Problem solved.