I have listened carefully, and my Gaelic is quite good. There is definitely no am there. True, it might disappear as the previous word ends in an a and the following word begins with an m, but that is the point - it has disappeared. With no other clues (such as the noun leniting if it were feminine) and with both sentences equally valid, there is simply no way to get it right in a tap what you hear question.
I agree, it is hard to hear the 'am' but I don't think it's the recording's fault necessarily. 'tha' ends in the first sound of 'am' and 'mions' starts with the last sound of 'am', so it is just difficult to discriminate. Probably in the context of a conversation (rather than an isolated sentence) and with familiarity with the language it would be easier to know whether the 'am' is in a sentence like this or not.
I agree with PurpleJulie26. This would not be a problem in a real-life situation. Either it wouldn't matter or you would find some way to make it clear - perhaps by saying
Tha am mions seo ceart gu leòr 'This mince is OK'
Tha mions sam bith ceart gu leòr 'Any mince is OK'
To show how similar they are, am mions can also mean 'their mince', but in Irish this would be (if they had the word mions which they don't) *a mions. Through a complicated process taking thousands of years, and not relevant to a Gaelic course, the m of am has been 100% absorbed into the mince - not heard and not seen.