Cardiac Magnetic Resonance (CMR) imaging is a non-invasive method for assessing cardiac structure, function, and blood flow. Cine MRI extends this by capturing heart motion, providing detailed insights into cardiac mechanics. To reduce scan time and breath-hold discomfort, fast acquisition techniques have been utilized at the cost of lowering image quality. Recently, Implicit Neural Representation (INR) methods have shown promise in unsupervised reconstruction by learning coordinate-to-value mappings from undersampled data, enabling high-quality image recovery. However, current existing INR methods primarily focus on using coordinate-based positional embeddings to learn the mapping, while overlooking the feature representations of the target point and its neighboring context. In this work, we propose KP-INR, a dual-branch INR method operating in k-space for cardiac cine MRI reconstruction: one branch processes the positional embedding of k-space coordinates, while the other learns from local multi-scale k-space feature representations at those coordinates. By enabling cross-branch interaction and approximating the target k-space values from both branches, KP-INR can achieve strong performance on challenging Cartesian k-space data. Experiments on the CMRxRecon2024 dataset confirms its improved performance over baseline models and highlights its potential in this field.