There are different situations where lenition and eclipsis are used. You will see fuller information in the "Tips and notes" for those skills, but here are two relevant examples for your question:
- Lenition is used after the possessive adjectives my, your (singular), and his: mo phláta, do phláta, a phláta.
- Eclipsis is used after the phrase ar an (on the), for example ar an bpláta. It is also used after a lot of similar phrases where you have a preposition followed by the word an: ag an bpláta, leis an bpláta, roimh an bpláta..
Irish "r"s have more flap to them than English ones. This makes it sound like an English d, which has a bit more tongue to it in Irish.
I believe another point of confusion could be that in some places "ar" (on) is pronounced like "air" (on him, said like "eh" followed by a slender "r", even though "ar" is written as broad). Micheál Ó Siadhail's "Learning Irish" gives it that way, and it sounds like how I heard it from my father (Galway). It sounds between "ed", a French "eh, j'...", or perhaps a very fast "edge", with the tongue flapping once rather than resting.
Pointed this out on a similar question in this set where the same mistake is made, but it sounds in the recording like she's saying "ar a bpláta" which means "on their plate". Is their a particular reason why the n in "an" isn't being enunciated before bpláta (especially since it very clearly is earlier in the sentence)?
I'm sorry, but elision is a normal part of ordinary speech, in both Irish and English (and presumably many other languages). That doesn't mean that you must elide at all times to be "correct".