Lowering Prescription Drug Costs and Extending Community Health Centers and Other Public Health Priorities Act This bill addresses prescription drugs and extends several health care, research, and workforce programs. Currently, a generic drug manufacturer is awarded 180 days of exclusivity on the market if it is the first applicant to file an application with the Food and Drug Administration (FDA), and the exclusivity period begins upon the first applicant's commercial marketing. The bill authorizes the FDA to approve a subsequent generic drug application prior to the first applicant's first date of commercial marketing if certain conditions are met. The bill prohibits a brand-name, generic, or biosimilar drug manufacturer from entering into certain agreements to resolve or settle a patent infringement claim in connection with the sale of a drug or biological product. Such an agreement, with specified exclusions, is a violation of the bill if the filer of the subsequent generic application receives something of value and agrees to limit or forego research, development, manufacturing, marketing, or sales of the generic drug or biological product. The developer of a drug or biological product, such as a generic or biosimilar version of a drug, may bring a civil action against the license holder of an approved drug if the holder has declined to make available sufficient quantities of the approved drug for the developer's testing. The bill also extends several health care, research, and workforce programs, including the National Health Service Corps, through FY2020.