Laser Direct Energy Deposition (LDED) emerges as a promising technology to repair damaged single crystal (SX) superalloy components with superior properties due to its high efficiency and good precision. However, stray grains (SGs) are still the major challenges that compromise repair quality though multiple techniques have been explored. The current work delves into understanding and mitigating SGs by physically elucidating the Parameter-Process-Structure (PPS) relationships. The Parameter-Process relationship is revealed by discussing the semicircular-to-undulating shape transition (SUT) under various parameter sets, probing the natural tendency of SX superalloy towards SGs through dimensional analysis and modeling the parameter-induced transitions in flow dynamics. The Process-Structure correlation is next derived by researching the oriented-to-misoriented transition (OMT) through transient thermal analysis regarding solidification and predicting the SGs fraction using high-fidelity simulation. The proposed PPS relationships in laser remelting are further confirmed in LDED by physically connecting the correlations between parameter variation-melt pool dynamics-SGs formation. It is revealed the melt pool of SX superalloy naturally shows susceptibilities of undulating shape and its boundary is controlled by thermal advection. The undulation is prone to being formed under higher heat energy input as the enhanced flow instability and the increased flow intensity characterized by the number and the intensity of inside vortex. SGs are sensitive to the undulating melt pool due to the inflection-induced increment in solidification angle and SGs fraction shows a significant increase with energy input. The obtained PPS physics work for both the laser remelting and LDED though the process complexity has been greatly raised by the powder stream and provide insights into the parameter optimization, process adjustment, and quality improvement for the laser repairing of SX superalloy-manufactured parts.