That is called 'intrusive r' and it's most notably an RP phenomenon, although not all RP speakers do this. Also, some speakers of American English (mostly in the north-east part of the States), Australian English and New Zealand English do have the 'intrusive r' as well. At least, if your aiming towards a British pronunciation, namely RP, it is up to you if you do it (I don't do it).
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askshameerMy instructor says that when a word ends with a vowel sound and if the next word starts with a vowel sound, then I'm supposed to add an "r" in between to read that out.This does not apply to American English in any significant way. If you are striving for an American accent (but it seems that you are not), don't follow that advice.