Pronunciation difference between "cycle" and "psycho"

The second syllable of cycle has the syllabic consonant (=acts like a vowel) l. Your tongue should be the front of the palate at the alveolar ridge, perhaps extending a bit to the teeth. You hold it there and create the syllabic center, even though it is traditionally classed as a consonant.

The second syllable of psycho is quite different, in that o it is a rounded vowel, and your tongue should not be touching your mouth anywhere.

Though I love tchrist's answer regarding the mechanics of pronouncing the words differently, I think that the fact is simply: there's an "L" in cycle. If you don't pronounce the "L", for whatever reason - cultural, lazy style, regional dialect - you will always stand the chance of being misunderstood, particularly by native speakers.

You mention "American English", so I suspect you are having a difficulty with the 'dark L'.

There are two (or more) L-sounds in (many varieties) of English: clear L and dark L.

Clear L is pronounced much like it is in French, Spanish, or German: /l/ in IPA. In American English it's usually found as initial L in words like leaf for example.

Dark L, /ɫ/ in IPA, is pronounced with the tip of the tongue touching the roof of the mouth behind the teeth, with the back of the tongue retracted back toward the throat. The sound is very much like /w/. In American English it's often found as final L in words like full or bottle.

You might be confusing /ɫ/ with /w/ in your speech. The way to avoid this is to ensure that in /ɫ/ the tip of your tongue touches the ridge behind your teeth.

Note: the distribution of the clear vs. dark L in various dialects of English is pretty complex so it's not surprising you may have a problem.