This is not a "tener que" expression! If anything comes between the words "tener" (conjugated) and "que," the sense of obligation, the "have to," is broken.
Consider some of the similar sentences Duo has already given us:
No tengo nada que añadir = I have nothing to add, not I have to add nothing!
Ella tiene algo que decirle a ella = She has something to tell/say to her, not She has to tell her something!
No tienes nada que perder = You have nothing to lose, not You have to lose nothing!
If that doesn't convince you, would you translate "No tengo nada que hacer" as "I have nothing to do" or as "I have to do nothing"? And, would "Tenemos bastante cerveza que beber" be "We have enough beer to drink" or "We have to drink enough beer"?
REVISED: The "que" in this sentence is a relative pronoun. It explains what we have a lot of. "Que" usually remains untranslated, however you could translate it as "which." "We have a lot (which) to do." Although I doubt that Duo would accept the "which."
This sentence seems to me to be a little ambiguous in the English translation given. The verb phrase "tener que" translates as "to have to," implying some level of obligation rather than just having a lot that could be done if we get around to it. I would argue the most accurate translation is, "We have to do a lot," rather than "We have a lot to do" (implying that we may or not get a lot done). Or maybe this distinction would not be made in Spanish???
Kikii, since you asked, "We have to do a lot" is wrong. You and your repliers following are on the wrong track as far as what Duolingo is teaching, here. Sometines itbus necessary to leave one's own ideas behind in order to learn something. See tge comment by amble2lingo above for a solid explanation about why your sentence is wrong
REVISED: No, JanKovacic, putting something between "tener" and "que" changes the meaning. See my comments above. In this sentence, "que" is a relative pronoun, probably best translated as "which." It is followed by an explanation of what we have a lot of. "We have a lot (which) to do." However I don't think Duo would accept the "which."
No. None of these sentences are right. http://lema.rae.es/dpd/srv/search?id=Uv3zo00r8D6awkUI8u
REVISED: The "que" in this sentence is a relative pronoun. It is followed by an explanation of what we have a lot of. The "que" usually remains untranslated, however you could translate it as "which". "We have a lot (which) to do." Duo probably won't accept the "which," though.
"A" is always a preposition or a personal "a" (unless it refers to the letter of the alphabet) and isn't needed here to translate "to" because the "to" is already built into the verb infinitive "hacer." Don't confuse this with the "a" that follows a conjugated verb and precedes an infinitive such as in: "Vinimos a visitar (We came to visit).
Hope this helps.