Large Language Model agents excel at solving complex tasks through iterative reasoning and tool use, but typically depend on ultra-large, costly backbones. Existing distillation approaches train smaller students to imitate full teacher trajectories, yet reasoning and knowledge gaps between the teacher and student can cause compounding errors. We propose SCoRe, a student-centered framework in which the student generates training trajectories and the teacher corrects only the earliest error, producing training data matched to the student's ability and exposing specific weaknesses. The student is first fine-tuned on corrected trajectories. Subsequently, short-horizon reinforcement learning starts from the verified prefix preceding the earliest error, with target rewards assigned at that step. This design encourages autonomous problem-solving beyond imitation and enhances training stability. On 12 challenging benchmarks, a 7B-parameter student distilled with SCoRe matches the agentic performance of a 72B-parameter teacher.