I'd take your agitator out and clean the underside of the agitator and the bottom of the spin basket under where the agitator was.
Also, most washers have a "clothes guard" fastened to the top of the outer drain basket to prevent clothes from either getting out of the inner spin basket or rubbing against the underside of the top of the washing machine. Clean the underside of that clothes guard.
What you're probably dealing with is soap scum; the same stuff that accumulates in your bath tub. Soap is soluble in water because one end of the soap molecule is polar and has an apparant negative charge. There is therefore an attraction between that end of the soap molecule and polar water molecules, and so soap molecules dissolve in water. But, hardness in the water, like Calcium++ and Iron+ ions react with these soap molecules and neutralize the apparant negative charge on them, causing them to lose their solubility in water, and precipitate out as soap scum. So, soap scum is nothing more than soap that has lost it's solubility in water, which is why you find these white stains hard to remove.
You can't remove them with soap and water because soap scum isn't soluble in water, but anything that you use to clean a bathtub should remove these white marks, especially a phosphoric acid based bathroom or toilet bowl cleaner. Phosphoric acid is the active ingredient in most bathroom cleaners because it'll cut through soap scum like a knife without harming chrome plating. Higher concentrations of phosphoric acid are commonly found in toilet bowl cleaners.
The difference between a "soap" and a detergent is that soap is a natural product made from animal fat (but usually vegetable oils) and a strong alkali. If caustic soda (NaOH) is used to make the soap, the result is a hard bar soap, typical of the stuff sold in your local supermarket. If potassium hydroxide (KOH) is used to make the soap, the result is a soft soap, like those used in hand lotions or shaving creams. Detergents, on the other hand, are synthetic soaps, made from chemicals in a lab. Because detergents are synthetic, they can be formulated to not react with the hardness in water to form a soap scum. This is precisely why you normally get a soap scum in your bathtub and bathroom sink, but never in your kitchen sink. It's because you normally use soap in your bathroom and detergents in your kitchen sink.
From the above, one would expect you shouldn't get a soap scum in your washing machine either because you use detergents in it as well. However, there's nothing else I can think of that would leave a "WHITE" stain on clothes other than soap scum. Are you using bar soap to remove marks from the clothes before laundering them?
PS: You don't need to know the rest:
Our word "soap" almost certainly comes from the name of Mount Sopa" in Italy. During Roman times, Mount Sopa was THE place to sacrifice small animals to the Gods in exchange for favours or good fortune. Roman citizens would light a fire, kill and cremate the dead animal on a spit, and the fat would drip off the animal and mix with the ashes from the wood used to burn it, forming a crude form of soap. Roman women noticed that clothes washed in the streams flowing down Mount Sopa after a rain would come cleaner than usual, and the reason was undoubtedly due to the crude soap that was washed down the mountain from the sacrificial fires.
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