If it is raining on Succos is there any benefit in eating in a Succah covered with a (Schlack) tarp over eating in the house? (sources please)
Per the Machtzis HaShekel Siman 640:9 it is preferable to eat in a Sukkah with a Schlack, over eating in the house when it is raining since there are Poskim that consider it a Kosher Succah.
The Bikurrei Yaakov Siman 626:12 says also that it is preferable to eat in a Sukkah with a Schlack, over eating in the house when it is raining.
Nope. If the s'chach isn't valid, it's not a sukkah.
The practical advantage is, you can have all sorts of nice stuff in the sukkah; if it starts to rain, just put up the tarp; as soon as it stops raining you can pull off the tarp and get right back to your mitzva. Whereas if you didn't have the tarp, you have to rush everything out of your sukkah when it starts to rain; then when the rain stops, you have to wait another hour+ for everything to dry out, then drag everything back into the sukkah. (Practically, people are likely to not bring out their best dishes, chairs etc. anyhow because of this.)
There is a ma'ala in sitting outside, even if the succah is covered. This is in accordance with the shita of the Rabbeinu Tam on maseches succah 10a SV 'piress'. The Shulchan Aruch (OC 629:19) only brings this down as a yesh omrim, but the MB says to be machmir to sit in the succah (without a brocha), b'shaas ha'd'chak, in other words, if you live in the UK like I do where it's almost always raining, if we were to say ha'mitztaer min ha'succah potur - we'd never be mekayem the mitzvah!