Protein search for specific binding sites on DNA is a fundamental biological phenomenon associated with the beginning of most major biological processes. It is frequently found that proteins find and recognize their specific targets quickly and efficiently despite the complex nature of protein-DNA interactions in living cells. Although significant experimental and theoretical efforts were made in recent years, the mechanisms of these processes remain not well-clarified. We present a theoretical study of the protein target search dynamics in the presence of semispecific binding sites which are viewed as traps. Our theoretical approach employs a discrete-state stochastic method that accounts for the most important physical and chemical processes in the system. It also leads to a full analytical description for all dynamic properties of the protein search. It is found that the presence of traps can significantly modify the protein search dynamics. This effect depends on the spatial positions of the targets and traps, on distances between them, on the average sliding length of the protein along the DNA, and on the total length of DNA. Theoretical predictions are discussed using simple physical-chemical arguments, and they are also validated with extensive Monte Carlo computer simulations.