Damages to Plug and or Cord exposes risk to property and people. House fires or electrocution could be a consequence of such a fault. The extensive use of powerboards and double adapters in a property can cause serious damage by overheating cables and contacts, worst case causing electrical fires and electrocution. A much safer option would be installing multiple power points on the wall.
A replacement of any damaged or compromised part is compulsory.
Here are some examples of non compliant and dangerous electrical parts.