"On Thursday an employee burns his hand whilst hot-working, the burn is very small and percieved as insignificant" - record the incident in your log/accident book or format that you use.

Review your risk assessments and control measures as you say "because burns and hot-working are very common" to ensure they are suitable and sufficient and ALARP based on a recorded incident.

"The employee works all Thursday and Friday without reporting the small burn to his employer." - tool box talk or similar making employees aware of their duties to report incidents and accidents promptly.

"Over the weekend the burn / open blister becomes infected (possibly due to the employee's outdoor hobbies, maybe also due to his burn aftercare) this eventually leads to his hand swelling up on the Sunday night and he is admitted to A&E and given anti-biotics." - ensure employees are educated on the best practice for care of burns post incident highlighting the benefits of proper care + getting better quicker and being able to get back to work and being paid (not sick pay etc.)