Very low pressure when turn on the hot water faucet in sink

kohler bancroft is the sink manufacturer, not the faucet mfgr. do you have a clue what faucet mfgr it might be?

The designs of the Kitchen Sink Faucet mainly exhibit differences in its handles and spouts. Differently curved spouts and handles diversify the range of products. Some have long spouts and some are with short ones. The spouting of water may also give different flow patterns with the external fixtures. The handles are mainly in single or double.


source Best Kitchen Faucets

You have already checked the two obvious things now make sure the two supply lines from the shut off valves are not kinked or turned it such away as to close off the water flow. If that is ok then there may be some debris in the internal filter screens.

Keep in mind that this faucet design will not put water out at a high pressure if it did it would splash all over the place.

If you have full pressure on the cold water side and low pressure on the hot side then you well need to remove and clean or replace the cartridge for the hot side. If you have low water pressure on both sides try removing the aerator and cleaning it.
If you need to replace the cartridge on the hot side do both sides while you at it.
Whenever you work on any faucet before you turn the water on remove the aerator and run the water for a few seconds then replace the aerator.

If I understand correctly, you have single handled shower faucet.
Normally there is a wait for hot water to arrive at faucet.
But now hot water is already at shower faucet.

Single handled faucets have a mixing area inside valve where hot and cold come together before going into sink.
This mixing valve can have crossover where the hot and cold can pass through to each other even when faucet handle is off. When single handle faucets get old, this can happen.
How it happens:
For example another faucet is turned on inside house, and pressure differential between hot and cold causes hot water at shower handle to pass through mixing area and into cold water line.
So if another faucet has been run, then that lets hot flow to faulty single handled shower faucet.
To confirm this is the problem: turn on other faucet and put ear against shower faucet to listen for moving water.
This condition may also signal there is a water leak.
The problem is not harmful, but will draw cold water into water heater each time, and is energy wasteful.
Replace faucet or have plumber repair problem.

bad faucet...get new at lowes or home depot

start by making sure the valve that supplies hot water to the sink and dish washer is fully open. If there is low hot water pressure in any of the other kitchen/bathrooms, then trace the line to see where the pressure drops. If it is only in the kitchen, it should be an old valve, (causes restriction)Good Luck.

it sounds like your cartrage is leaking or it's the connection between the handle and the spout

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.

It could be one of two problems, one simple, one serious.

• Easy fix
  

1. Turn off hot water angle stop under sink to faucet.
2. Remove supply line from hot angle stop.
3. Place end of supply line in a small bucket to catch some water.
4. Turn on cold water for a couple of seconds and backflush debris from faucet.
5. Reconnect supply line.
6. Turn on angle stop and test
   

• Complex fix

If you have steel pipes you may be experiencing reduced flow due to corrosion in your pipes. Reduced hot water pressure in the kitchen faucet is usually the first sign since this is the faucet that is most used in the home.

Repiping is a job for a professional. Another symptom you may need a repipe is orange colored or rusty water.