We give improved deterministic algorithms solving sparse instances of MAX-SAT and MAX-k-CSP. For instances with n variables and cn clauses (constraints), we give algorithms running in time \poly(n)\cdot 2^{n(1-\mu)} for
\begin{itemize}
\item \mu = \Omega(\frac{1}{c} ) and polynomial space solving MAX-SAT and MAX-k-SAT,
\item \mu = \Omega(\frac{1}{\sqrt{c}} ) and exponential space solving MAX-SAT and MAX-k-SAT,
\item \mu = \Omega(\frac{1}{ck^2} ) and polynomial space solving MAX-k-CSP,
\item \mu = \Omega(\frac{1}{\sqrt{ck^3}} ) and exponential space solving MAX-k-CSP.
\end{itemize}
The previous MAX-SAT algorithms have savings \mu=\Omega(\frac{1}{c^2 \log^2 c}) for running in polynomial space~\cite{SST14} and \mu=\Omega(\frac{1}{c \log c}) for exponential space~\cite{DW06}. We also give an algorithm with improved savings for satisfiability of depth-2 threshold circuits with cn wires.