Incomplete-Modality Multimodal Sentiment Analysis extends multimodal sentiment analysis (MSA) to infer sentiment from language, audio, and visual streams when one or more modalities are incomplete at inference time. Existing methods improve incomplete-modality MSA through feature reconstruction, distillation, or fusion adjustment, but usually assign similar correction strength to all samples with incomplete inputs. This overlooks the uneven emotional impact of modality missingness: mild missingness often leaves sentiment recognizable, whereas severe missingness can distort the expressed emotion and therefore requires stronger correction. We propose HyDRA (Hyperbolic Damage-aware Recovery and Adaptation), a damage-aware and text-guided hyperbolic framework that aligns correction strength with missingness-induced sentiment damage. HyDRA first recovers lost emotional cues from available modalities, then places each sample in the Poincaré ball according to estimated damage severity, and finally applies damage-weighted hyperbolic correction toward a complete-modality teacher anchor. Since hyperbolic distances expand near the boundary, heavily incomplete samples receive stronger correction while lightly incomplete samples are gently adjusted. Experiments on MOSI, MOSEI, and SIMS show SOTA robustness across incomplete-modality conditions.