A bad power steering pump cannot cause loose steering but it can make it difficult to steer the vehicle and the steering will be harder to turn.
When your steering is loose the most common cause of loose steering is bad tie rods, bad joints, bad rack and pinion, bad steering linkages, bad steering arms, bad pitman arms, or bad suspension.
The life expectancy of a power steering pump is around 100,000 miles and sometimes longer or around 10 years or more depending on how much you drive.
The symptoms of a failing power steering pump are.
A squealing sound when you start the car.
Groaning noises when you turn the steering wheel.
Steering wheel that is slow to respond.
Stiff steering wheel.
Low power steering fluid and leaking power steering fluid.
And grey power steering fluid.
If your power steering pulley is not spinning then the power steering pump has likely seized up and needs replaced.
The power steering pump pulley should spin with a slight drag when spun by hand and if it's really tight or it won't spin at all then you need a new power steering pump.
A power steering pump should turn freely and the shaft should free spin easily in the housing.
If the power steering pump pulley is hard to turn or the power steering pump shaft is hard to spin then it could be a sign that the power steering pump is going bad.
The pulley on the power steering pump should fit tightly on the shaft and not spin around on the shaft or be loose.
Power steering pump pulleys are pressed on and require a power steering pump pulley installer tool and removal tool.
Never try to pry or hammer on the power steering pump as it can damage the power steering pump.
Also never use a jaw type puller such as a gear puller on a power steering pump pulley as it can damage it.
But if replacing the power steering pump you should always use a new pulley so that it presses on tightly.
Without the friction of the drive belt and the engine NOT running, all driven pulleys should turn freely without noise when spinning by hand.
The P/S pump and water pump pulley will have some drag (resistance) but still turn freely with little resistance when turned relatively slowly.