Translation:I sing, you play music, and Péter washes the dishes.
Unfortunately it happens often in spoken language, too, even some dialects drop the L in such positions by rule. As the audio track is recorded, the living speech may have such things even if it is purpose-made by an actress (AFAIK) and her pronounce is very accurate. I can hear the L there—but only if I want it really hard.
In a case like this, "zenész" (=musician) would render a really strange meaning (rather a kind of nonsense), so you may assume the L there...
"Mosogat", "elmosogat", "elmossa az edényeket", "elmossa a ...-t"
I can't find a Hungarian phrase for this action where "washing" doesn't appear. In Hungarian you don't do/csinálod the dishes, unless you're actually making them.
If you asked this because "doing the dishes" was not accepted, report it please.
I don't see real reason for that, but the meaning of the sentence would change if you use "és" here instead of "pedig". Both version could appear as an order or a statement on the actual actions. But the "és" version is more on the order side. For using as a statement on actual action it expresses a bit of surprised overtone for me as you expected Peter to sing and "you" to do the dishes.