Is the pronunciation here incorrect or are you supposed to barely here an 'a' after the 'e'?
It is correct. But you ar supposed to hear a demivowel (I don't really know if that exists in English. I think it would be similar to a short vowel) and a is the longer vowel.
DL dropdown menu shows "wish" as an option. But, "wish = "dori" and "veau" = "want".
The verbs "a dori" and "a vrea" have the same meaning, with the former more formal than the latter.
However, please keep in mind that "a dori" has, in addition, a reflexive form in the dative case, meaning "to wish for": Eu îmi doresc; Tu îți dorești; El/ea își dorește; Noi ne dorim; Voi vă doriți; Ei/ele își doresc. There is no equivalent in this respect for "a vrea".