We investigate exceptional sets associated with various additive problems involving sums of cubes. By developing a method wherein an exponential sum over the set of exceptions is employed explicitly within the Hardy-Littlewood method, we are better able to exploit excess variables. By way of illustration, we show that the number of odd integers not divisible by $9$, and not exceeding $X$, that fail to have a representation as the sum of $7$ cubes of prime numbers, is $O(X^{23/36+\eps})$. For sums of eight cubes of prime numbers, the corresponding number of exceptional integers is $O(X^{11/36+\eps})$.

Let $b_1,\dots,b_5$ be non-zero integers and $n$ any integer. Suppose that $b_1 + \cdots + b_5 \equiv n \pmod{24}$ and $(b_i,b_j) = 1$ for $1 \leq i < j \leq 5$. In this paper we prove that \begin{enumerate}[(ii)] \item[(i)] if $b_j$ are not all of the same sign, then the above quadratic equation has prime solutions satisfying $p_j \ll \sqrt{|n|} + \max \{|b_j|\}^{20+\ve}$; and \item[(ii)] if all $b_j$ are positive and $n \gg \max \{|b_j|\}^{41+ \ve}$, then the quadratic equation $b_1 p_1^2 + \cdots + b_5 p_5^2 = n$ is soluble in primes $p_j$. \end{enumerate}