In the cosmic history, studying the later part of the cosmic evolution, in particular, large-scale structure formation is the key in understanding a plethora of new and exotic objects in the Universe. Recent advancements in the multi-frequency observations of low-redshift galaxy clusters has allowed us to have a considerable amount of insight into the physical processes occurring in these objects. Although much is known about the extended radio emission from the clusters, with the size spanning between ~0.3–1.5 Mpc, we are far from having a good understanding of the nature and origin of these objects. These diffuse radio objects are broadly classified as Halos, Relics and minihalos. However, there are also a large number of morphologically complex and/or intermediate class objects found in recent surveys. But for deeper understanding of these objects and their role in cosmic evolution, we need more sample across large redshirt range with different physical properties like mass, X-ray luminosity etc. that are studied in multiple frequencies. Here, we have presented a study of a few of them with radio and X-ray observations. Firstly, with 610 MHz and 1.5 GHz radio observations, we have estimated the spectral index of the most distant minihalo in the Universe, the Phoenix cluster, to be -0.98. Furthermore, Chandra X-ray observations of this cluster has revealed a spiralling cool gas around the cluster core along with two cold fronts, which implies the presence of gas sloshing in the intra-cluster medium (ICM). Following the evidences in the literature, we infer that sloshing has injected turbulence in the ICM which is re-accelerating the in situ seed electrons, probably coming from the central AGN, resulting in the observed minihalo in this cluster. Secondly, we discovered a new diffuse radio object in the SPT-CL J2031-4037 cluster with 325 MHz and 1.7 GHz observations. The dynamical state parameters suggest it being a less-energetic intermediate state merger. The spectral index distribution was found to be steep (~-1.35) along the merger axis and ultra-steep in the off-axis region. The size of the diffuse radio emission as well as the cluster morphology suggest it being an intermediate class halo. Lastly, a low-frequency (325 MHz) radio observation campaign of 15 SPT (South Pole Telescope) clusters has resulted in the discovery of 2 new radio halos, 2 literature halos along with a candidate halo. These studies has contributed in understanding the diffuse radio emission from galaxy clusters and also demonstrated the necessity for further rigorous investigation of these radio sources.