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...
You made me realize we basically have two different 'l's. One is when the tip of our tongue touches our front teeth, and the other is when we avoid that and instead curl the sides of our tongue up.
Doing the washing up wasn't accepted, but is what one would say in English.
In english -doing- the dishes means -washing- the dishes. Is there a difference in hungarian?
"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.