The correct option is
B perspiration
The words, 'drenched with something" means 'to be thoroughly wet with something, usually a liquid'. Of the given options, the word, 'perspiration' refers to the process of sweating. Hence, the clause, 'my back was drenched with perspiration' means 'my back was thoroughly wet with sweat drops'. Hence, option B is most appropriate.
Option A: The word, 'prickly heat' refers to an itchy inflammation of the skin, which is common in hot, humid weather.
Option B: The word, 'sores' refers to raw or painful marks on the body.
Option D: The word, 'fatigue' refers to extreme tiredness resulting from mental or physical exertion or illness.
As the words, 'prickly heat', 'sores' and 'fatigue' do not refer to liquids, 'drenched with prickly heat/sores/fatigue' are incorrect usages. Hence, options A, C and D are incorrect.