Conceding Our Veterans' Exposures Now And Necessitating Training Act of 2021 or the COVENANT Act of 2021 This bill establishes a presumption of service-connection for certain illnesses related to exposure to airborne hazards or open burn pits and addresses the care available through the Department of Veterans Affairs (VA) for veterans with illnesses they assert to be related to exposures during their service. Specifically, the bill establishes a presumption of service-connection for certain illnesses becoming manifest in covered veterans to a degree of 10% or more. Under a presumption of service-connection, specific conditions diagnosed in certain veterans are presumed to have been caused by the circumstances of their military service. Health care benefits and disability compensation may then be awarded. A covered veteran is any veteran who served on or after August 2, 1990, in Bahrain, Iraq, Kuwait, Oman, Qatar, Saudi Arabia, Somalia, or the United Arab Emirates; or served on or after September 11, 2001, in Afghanistan, Djibouti, Egypt, Jordan, Lebanon, Syria, Yemen, Uzbekistan, the Philippines, or other countries the VA determines to be relevant. Among other elements, the bill requires the VA to provide medical examinations for veterans who submit a claim for disability compensation for an illness not listed in the bill but that is asserted to be related to airborne hazard or open burn pit exposure; authorizes the VA to stay pending claims for disability compensation related to the service and illnesses covered by this bill until the implementation of the bill; and provides VA hospital care, medical services, and nursing home care to covered veterans.