To what extent the fact of mispronouncing words such as "cup" , "cap", "mad", "mud", and so on...prevent you - English speakers - from understanding us?. I find those two phonems especially difficult to achieve. Could anyone give me/us-foreigners- a useful tip to deal with it successfully?
65 views in the last 24 hours, and no replies! I'm afraid this is a tricky one to answer. The sounds, in acoustic terms, are very close, but because they're phonemically distinctive for us it is easy for us to distinguish them; in fact, for a native speaker of English it's surprising to learn that they are so similar - 'How can that be...' we ask when shown the evidence (a spectrograph, say) '...when they sound so different?'
In actual cases, depending on context, getting it wrong can be either negligible/charming/amusing, or seriously misleading; it really does depend. If, for example, you're offered tea (well, I am English) and you say 'Just a small cap please', the error may not even register - you'll just sound a bit odd. If you go into a Police station and say 'Somebody stole my cap', they'll assume you mean headgear.
I'm glad someone else took a chance at attempting to help your predicament, and there probably isn't any better advice.
Really, the whole vowel system in phonetics is abstract. With modern imaging techniques, the whole 'vowel space' that Daniel Jones introduced with close, mid, open/front, central, back distinctions, we see that these distinctions are just too inaccurate.
The best way to learn vowel sounds is probably by listening and imitating. No amount of description would work - the movements are just too intricate and complicated, especially in connected speech.
Also, vowels are one of the main distinguishing factors in different accents, and there are many different accents in native English speaking countries, as well as many people with English as a second language who bring their own accent to the language.
What I mean is people are very forgiving of accents, and thus of vowel idiosyncracies. It might cause some amusement. For example, in my own accent, I pronounce 'ice' the way many English speakers pronounce 'ace', and the way I pronounce words with an /aʊ/ diphthong in RP English causes great amusement for my fiancée, who speaks English with a Brazilian accent and says /aʊ/, in a sentence like 'how now brown cow'. Apart from amusement, it shouldn't cause too much bother.