Flights to and from smaller low-traffic airports are usually quite expensive because of lack of competition. But why do you want to fly to Leipzig when you are going to Erfurt? Most flights, at least the shortest ones, from Paris to Leipzig will have a transfer in Frankfurt and it is most likely just as fast to take the train from Frankfurt Airport to Erfurt (about 2:30 travel time and frequent departures) as to have an additional flight from Frankfurt to Leipzig and then go from Leipzig to Erfurt by train (about 1:00 travel time).
Considering the travel times, no matter what option you choose, I still don't understand why you want to make this a day trip.
Checking for an option on Saturday October 8th (since you don't reveal which date you are intending to travel and also seem to be slightly confused, asking about a Saturday in the text, but having searched yourself for options on a Wednesday), the first flight from Paris to Frankfurt arrives 9:40 and with about 1h time to transfer catch a train connection arriving in Erfurt 13:07.
Then, the last connection from Frankfurt back to Paris is departing 18:30 with a transfer in Amsterdam. To catch that flight, you will have to leave Erfurt 14:17 or if you risk a shorter time for checkin in Frankfurt, 14:50 may suffice. You will have something between 60 and 90 minutes to spend with your friend in Erfurt. If you book now, you can get the plane tickets for around 200€ and the train tickets for 50€. But does that make any sense at all?