The recently proposed Atomistic Structure Learning Algorithm (ASLA) builds on neural network enabled image recognition and reinforcement learning. It enables fully autonomous structure determination when used in combination with a first-principles total energy calculator, e.g. a density functional theory (DFT) program. To save on the computational requirements, ASLA utilizes the DFT program in a single-point mode, i.e. without allowing for relaxation of the structural candidates according to the force information at the DFT level. In this work, we augment ASLA to establish a surrogate energy model concurrently with its structure search. This enables approximative but computationally cheap relaxation of the structural candidates before the single-point energy evaluation with the computationally expensive DFT program. We demonstrate a significantly increased performance of ASLA for building benzene while utilizing a surrogate energy landscape. Further we apply this model-enhanced ASLA in a thorough investigation of the c(4x8) phase of the Ag(111) surface oxide. ASLA successfully identifies a surface reconstruction which has previously only been guessed on the basis of scanning tunnelling microscopy images.