For \(1\leq k\in {\mathbb{R}}\), \(r\in {\mathbb{N}}\) let \(t_ k(r):=\#\{(m,n)\in {\mathbb{N}}\times {\mathbb{Z}}:\) \(m^ k-| n|^ k=r\}\). The authors generalize and supersede an asymptotic result of E. Krätzel [Acta Arith. 16, 111-121 (1969; Zbl 0201.050)] concerning the average order of \(t_ k\) ( ) for 3-1/13\(\leq k\in {\mathbb{R}}\) instead of \(3\leq k\in {\mathbb{N}}\). Let \(T_ k(x)\) denote the corresponding summatory function, then it is proved that \[ T_ k(x)=c_ 1(k)x^{2/k}+c_ 2(k)x^{1/(k-1)}+c_ 3(k;x)x^{1/k-1/k^ 2}+D_ k(x) \] with explicit constants \(c_{1,2}(k)\), \(c_ 3(k;x)=O(1)\), \(\Omega_{\pm}(1)\) for \(x\to \infty\) (well-known) and \[ \Delta_ k(x)=O(x^{2/(3k)-1/(114k)+\epsilon}) \] for any \(\epsilon >0\). The key idea here is a skilful splitting-up argument in the representation of \(\Delta_ k( )\) in terms of \(\psi_ 1\)-sums, so that the method of (one-dimensional) exponent pairs pushed by M. N. Huxley and N. Watt is now applicable to this problem. Note that \(T_ 2( )\) is intimately related with Dirichlet’s divisor problem [W. Sierpinski, Wiadom. Mat. 11, 89-110 (1907)].