Tailoring motor activities to the unique needs of children with special educational requirements has shown considerable efficacy. Our study aimed to develop a structured program specifically designed to enhance psychomotor abilities, focusing on balance and motor–cognitive skills among 28 students (aged 12–14) from two institutional centers in Romania. The program spanned 36 weeks, with biweekly 30 min sessions. Psychomotor skills were assessed through tests measuring balance, speed of movements, and upper limb motor laterality. Initial and final data were collected for evaluation. A statistical analysis, employing the Kolmogorov–Smirnov and Wilcoxon Z tests, compared the assessments. The results indicated significant improvements in movement speed, with a notable increase in stimulus identification (averaging from approximately 13 to 14) (p < 0.05) and reduced processing time (decreasing from about 28.7 to 28 s) (p < 0.05). However, while the structured program demonstrated substantial enhancements in specific motor and cognitive–motor skills, it did not yield significant changes in dynamic balance, maintaining values close to 0.9 (p > 0.05) for open-eyed balance and 0.88 (p > 0.05) for closed-eyed balance. Additionally, an analysis of the processing speed in pulses per second showcased a marginal decline, from approximately 0.46 to 0.45, revealing notable disparities between the initial and final measurements (p < 0.05).