This study introduces an automatic assessment model for speech production skills of children with cochlear implants (CIs) to support home-based speech therapy. The model employs acoustic embeddings from self-supervised models and considers speech traits of both normal hearing (NH) adults and children, which is a novel method for evaluating speech of children with disorders. It combines phoneme embeddings and two acoustic embeddings from Wav2Vec2.0 models, each trained on the speech of NH adults and children, via multi-head attention. Using a speech corpus of Korean-speaking children with CIs, our model outperforms single-embedding methods in a Pearson correlation coefficient between predicted and expert-rated scores, with a relative improvement of 51%. The results highlight the effectiveness of Wav2Vec2.0 acoustic embeddings and the importance of incorporating both of typical speech patterns of NH adults and children in assessing speech production skills in children with CIs.