I need help knowing what $cpm \times 10^3$ means in Figure 4(C) of this paper (https://www.sciencedirect.com/science/article/pii/S0022202X15323149#f0010). It appears to be a unit of kinase activity.
It is not a specific unit of kinase activity, rather a general approach taken for following reactions where there is no change of colour. The abbreviation cpm stands for counts per minute and is a measure of radioactivity. In this particular example a radioactive phosphate has been incorporated in the ATP of the protein kinase assay. If the protein kinase is active with the substrate, the phosphate will be transferred from the ATP to the substrate peptide/protein. If the sample mix is loaded on a SDS-PAGE gel, a scintillation counter can be used to measure the radioactivity of each band and one will be able to show how active the protein kinase has been in phosphorylating it's substrate.