Purpose: To develop a scan-specific model that estimates and corrects k-space errors made when reconstructing accelerated Magnetic Resonance Imaging (MRI) data. Methods: Scan-Specific Artifact Reduction in k-space (SPARK) trains a convolutional-neural-network to estimate and correct k-space errors made by an input reconstruction technique by back-propagating from the mean-squared-error loss between an auto-calibration signal (ACS) and the input technique's reconstructed ACS. First, SPARK is applied to GRAPPA and demonstrates improved robustness over other scan-specific models, such as RAKI and residual-RAKI. Subsequent experiments demonstrate that SPARK synergizes with residual-RAKI to improve reconstruction performance. SPARK also improves reconstruction quality when applied to advanced acquisition and reconstruction techniques like 2D virtual coil (VC-) GRAPPA, 2D LORAKS, 3D GRAPPA without an integrated ACS region, and 2D/3D wave-encoded images. Results: SPARK yields 1.5x - 2x RMSE reduction when applied to GRAPPA and improves robustness to ACS size for various acceleration rates in comparison to other scan-specific techniques. When applied to advanced reconstruction techniques such as residual-RAKI, 2D VC-GRAPPA and LORAKS, SPARK achieves up to 20% RMSE improvement. SPARK with 3D GRAPPA also improves performance by ~2x and perceived image quality without a fully sampled ACS region. Finally, SPARK synergizes with non-cartesian 2D and 3D wave-encoding imaging by reducing RMSE between 20-25% and providing qualitative improvements. Conclusion: SPARK synergizes with physics-based acquisition and reconstruction techniques to improve accelerated MRI by training scan-specific models to estimate and correct reconstruction errors in k-space.