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The paper deals with very weak solutions $u\in \theta +W_0^{1,r}\left(\Omega \right)$, $max\{1,p-1\}<r<p<n$, to boundary value problems of the $p$-harmonic equation $$\left\{\begin{array}{cc}-\text{div}\left(\right|\nabla u\left(x\right){|}^{p-2}\nabla u\left(x\right))=0,\hfill & x\in \Omega ,\hfill \\ u\left(x\right)=\theta \left(x\right),\hfill & x\in \partial \Omega .\hfill \end{array}\right.(*)$$ We show that, under the assumption $\theta \in W^{1,q}\left(\Omega \right)$, $q>r$, any very weak solution $u$ to the boundary value problem ($*$) is integrable with $$u\in \left\{\begin{array}{cc}\theta +L_{\mathrm{weak}}^{q^{*}}\left(\Omega \right)\hfill & \text{for}q<n,\hfill \\ \theta +L_{\mathrm{weak}}^{\tau }\left(\Omega \right)\hfill & \text{for}q=n\text{and}\text{any}\tau <\infty ,\hfill \\ \theta +L^{\infty }\left(\Omega \right)\hfill & \text{for}q>n,\hfill \end{array}\right.$$ provided that $r$ is sufficiently close to $p$.