Are SAT problems with at most one false clause NP-complete?
Given a CNF with $n$ variables and $m$ clauses, we denote by $A_i$ the set of truth assignments that do not satisfy clause $i$. Then the number of satisfiable assignments is 
$$2^n-\left|\bigcup_{i=1}^m A_i\right|.$$
Since each assignment makes at most one clause false, $A_i$ and $A_j$ are disjoint for $i\neq j$. Also note $|A_i|=2^{n-n_i}$ where $n_i$ is the number of variables in clause $i$. Hence, the number of satisfiable assignments is exactly $2^n-\sum_{i=1}^m 2^{n-n_i}$ (this result is also given as lemma 2 in [1]). 

So we can decide whether the CNF is satisfiable by checking whether $2^n-\sum_{i=1}^m 2^{n-n_i}&gt;0$, which can be done in polynomial time.



[1] Nishimura, N., Ragde, P., &amp; Szeider, S. (2007). Solving #SAT using vertex covers. Acta Informatica, 44(7-8), 509-523.