Private information retrieval has been reformulated in an information-theoretic perspective in recent years. The two most important parameters considered for a PIR scheme in a distributed storage system are the storage overhead and PIR rate. The complexity of the computations done by the servers for the various tasks of the distributed storage system is an important parameter in such systems which didn't get enough attention in PIR schemes. As a consequence, we take into consideration a third parameter, the access complexity of a PIR scheme, which characterizes the total amount of data to be accessed by the servers for responding to the queries throughout a PIR scheme. We use a general covering codes approach as the main tool for improving the access complexity. With a given amount of storage overhead, the ultimate objective is to characterize the tradeoff between the rate and access complexity of a PIR scheme. This covering codes approach raises a new interesting coding problem of generalized coverings similarly to the well-known generalized Hamming weights.