How do I tighten a single hole kitchen faucet to the sink. My faucet doesn't leak but moves around on the sink when I use it.

Sounds like the nut attaching faucet under sink has loosened. You have to crawl under sink to tighten it.

Shut off the water

Clean and dry the area around the faucet.

(from the bottom)
Disconnect/remove the faucet (clean or replace gasket).

Clean the hole completely & apply desired sealant

Positon the faucet in the hole & TIGHTEN (firmly).

Reconnect water lines & restore water supply.
(do NOT over-tighten).

Place clean "printer" paper under each wall valve & supervise
for several days to assure no DROPS are leaving wet marks on the paper.

Be proud.

Carnac the Magnificent

Check whether the faucet is leaking or the drainage is leaking.
If the faucet is leaking, you need tools to loosen the faucet to check whether the hoses are leaking or not. If the drainage is leaking, you can tighten the slip nut.

make this steps and fix it. God bless you
f a faucet leaks, replace seals and gaskets . If a shower head leaks, create a better seal or replace seals .
If a sillcock wears out, replace the washers .
If a compression faucet drips from the spout, replace the washer and/or the seat .
If a faucet leaks from the handle, replace O-ring(s), replace the packing washer, tighten the packing nut, or replace worn packing, depending on the valve type.
If a ball or cartridge faucet leaks from the collar, replace worn O-rings .
If a disk faucet leaks around the base or has reduced flow, replace cracked or pitted disk assembly or worn inlet seals
If there is water under the sink, tighten the faucet-set locknuts under the sink. You also can replace putty or a gasket, replace a worn faucet, or replace leaky supply tubes .
If the aerator leaks around the edge, replace the washer in the aerator
If the spray hose leaks or has reduced flow, replace the O-ring on the diverter valve, replace a worn washer at the base of the spray head, or clean the diverter valve and spray head

make this steps and fix it. God bless you
f a faucet leaks, replace seals and gaskets . If a shower head leaks, create a better seal or replace seals
If a sillcock wears out, replace the washers
If a compression faucet drips from the spout, replace the washer and/or the seat
If a faucet leaks from the handle, replace O-ring(s), replace the packing washer, tighten the packing nut, or replace worn packing, depending on the valve type
If a ball or cartridge faucet leaks from the collar, replace worn O-rings
If a disk faucet leaks around the base or has reduced flow, replace cracked or pitted disk assembly or worn inlet seals
If there is water under the sink, tighten the faucet-set locknuts under the sink. You also can replace putty or a gasket, replace a worn faucet, or replace leaky supply tubes
If the aerator leaks around the edge, replace the washer in the aerator
If the spray hose leaks or has reduced flow, replace the O-ring on the diverter valve, replace a worn washer at the base of the spray head, or clean the diverter valve and spray head

If you still have the receipts from the purchase call Moen directly as they warranty their faucets for life and will send you the replacement parts you need. If you need to purchase replacement parts then you can purchase a Moen single lever repair kit from your local hardware store which should run you around $10. Go the the Moen website and look for your model. They generally have an exploded view of every one of their models that will assist you in dissassembling the tap without any surprises. Reassemble the tap with the new o-rings etc installed and test.

you may need to get the moen cartridge puller,either moen brand, or brass craft, they can be tough to pull some time's

IF YOU ARE TELLING ME THE FAUCET ITSELF IS MOVING TO THE LEFT OR RIGHT THE FAUCET MAY NEED TIGHTENING
REMOVE THE SPOUT HANDLE, UNDER THE HANDLE YOU WILL SEE A PLASTIC NUT ASSEMBLY THAT YOU CAN TIGHTEN SEE IF THIS SOLVES THE PROBLEM

pull out the spray part and tighten the nut around the hose or take it loose and replace the washer

There is a bracket that slides between the threaded shaft and the bottom of the sink. There is usually a nut which is tightened up against this bracket. When this nut is loosened, the bracket will slide out. Of course, you first have to unfasten the two water supply lines. Once the bracket is removed, the two water lines which are made into the faucet are gently bent and the whole thing is removed from the top of the sink. Hope this helps.