What caused the Second Opium War (1856-60) was that Britain wanted to renegotiate the Nanking Treaty to improve its prerogatives, including the legalization of the opium trade, but the Qing authorities opposed this.
Consequently the british needed a new excuse to start a new war and they found it with the incident of the ship Arrow. This was anchored in Canton, when he was assaulted by the imperial troops and his chinese crew arrested under the accusation of smuggling. The british claimed that it was a ship with british flag. Even so, the british took it as an affront and bombarded Canton from the sea in the guise of retaliation. The Second Opium War, also known as the second Anglo Chinese War, had begun.
France joined the british wih another excuse. The death of a french missionary in Guangxi, executed by the local authorities for preaching the Christian religion in a province where he did not have permission ( only evangelize in Canton).