To understand what Paul meant by a "new lump" we need to recognize that ancient bread baking was slightly different than it is today in regard to how the bread would be leavened. Modern bakers have available to them packaged yeast which was not available back then. Bakers today will add a quantity of yeast to flour and water, the yeast will eat the carbohydrates and fart carbon dioxide. The farts rise through the dough causing it to rise. There is a similar process that goes on in the making of alcoholic beverages. Yeast eats the sugars in the fruit and produce alcohol.
Yeast occurs naturally in abundance in the air so if you set out a vat of grapes for example the yeast from the air will find the sugar and multiply and turn the juice to wine. And if you cultivate the yeast that eats the starches in bread dough your dough serves as a starter to make loaves of bread rise:
Mat_13:33 Another parable spake he unto them; The kingdom of heaven
is like unto leaven, which a woman took, and hid in three measures of
meal, till the whole was leavened.
Now, in the process of making bread one often lets the bread rise and then punches it back down to rise again.
Soooo, if you don't want your bread to be "puffed up" (as with pride) you don't allow the yeast time to multiply before you bake it resulting in a flat bread, like Pita bread. However, if it has already risen one has to punch it down to knock the air out of it. This is what I believe Paul is describing when he says to "purge out the old leaven to become a new lump". He's not saying, "get rid of those little packets of yeast" but rather "punch the air out so you can be an unpuffed up lump".