Diffusion MRI offers a unique probe into neural microstructure and connectivity in the developing brain. However, analysis of neonatal brain imaging data is complicated by inevitable subject motion, leading to a series of scattered slices that need to be aligned within and across diffusion-weighted contrasts. Here, we develop a reconstruction method for scattered slice multi-shell high angular resolution diffusion imaging (HARDI) data, jointly estimating an uncorrupted data representation and motion parameters at the slice or multiband excitation level. The reconstruction relies on data-driven representation of multi-shell HARDI data using a bespoke spherical harmonics and radial decomposition (SHARD), which avoids imposing model assumptions, thus facilitating to compare various microstructure imaging methods in the reconstructed output. Furthermore, the proposed framework integrates slice-level outlier rejection, distortion correction, and slice profile correction. We evaluate the method in the neonatal cohort of the developing Human Connectome Project (650 scans). Validation experiments demonstrate accurate slice-level motion correction across the age range and across the range of motion in the population. Results in the neonatal data show successful reconstruction even in severely motion-corrupted subjects. In addition, we illustrate how local tissue modelling can extract advanced microstructure features such as orientation distribution functions from the motion-corrected reconstructions.