Embedded or immersed boundary methods are powerful tools in computational fluid dynamics for solving partial differential equations in complex geometries, as generating body-fitted meshes is challenging and often results in intricate or irregular mesh shapes. Despite decades of significant research, automated processes capable of handling complex geometries are still lacking. This challenge becomes especially severe when the problem involves moving boundaries, as it requires remeshing at each time step, significantly increasing the computational cost of the simulation. Despite these advantages, embedded methods face challenges, particularly their limited accuracy in high-error regions such as boundary layers or vortex areas. This issue arises from their predominant use with Cartesian meshes. One way to address this limitation is by integrating adaptive mesh refinement strategies. Existing approaches, such as h-refinement (adding elements) and p-refinement (increasing polynomial order), improve accuracy but do not account for mesh alignment. Mesh alignment is crucial for computationally efficient adaptation in regions with strong gradients in one direction, such as boundary layers. Aligned high-aspect-ratio elements are especially important in high-Reynolds-number flows, where thin boundary layers require careful simulation. Addressing this, r-adaptation methods can align and concentrate the mesh near boundaries by deforming it. The effectiveness of the r-method in adapting the mesh to complex-shaped boundary layers has already been investigated in the literature [1]. In addition, h-refinement can increase resolution in regions with localized phenomena, such as vortices, without affecting neighboring elements. In this study, we consider r-adaptive methods and their combination with h-refinement approaches, creating an r-h refinement method, which is integrated into the embedded finite element method to enhance accuracy. The first step of the proposed r-h method involves deforming the mesh using an r-adaptive method around boundary layers, followed by a second step that applies h-refinement in local high-error regions during the simulation. Numerical tests demonstrate significant accuracy improvements compared to a uniform mesh embedded method in heat transfer and incompressible flow simulations.