I have a LG front loader Washer, and after a complete wash cycle, the detergent cup is filled with clear water.

Hi from retired Englishman in SW France,

This is a common problem with many machines and I suspect that the detergent drawer needs a thorough clean!
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Remove the complete drawer. To do this you will have to do one of the following-

• push down the lip/catch at the middle rear and pull the drawer out (that's how our Bosch one is removed)
• release the catch underneath or
• lift upwards on the tray as it is coming out to clear the lip underneath.

One of these should enable you remove it
(I am having to guess some of the following!)
Once out identify which tray is for the conditioner- the others are simply 'chutes' and open-ended.
It will normally have some sort of cover on it which you must remove to expose the little stuck up pipe/tube. This is down which the conditioner and water should siphon. It is likely to be blocked by 'gunge'- a mix of powder and conditioner. The whole of the tray, particularly this pipe and the bit which fits over it, must be completely clean and clear for it to operate correctly. It is worthwhile checking fairly regularly.
I hope this is your problem and solution!
Good luck,
John C