A contractor has to finish a contract of laying drainage

Actually, your way of thinking is faster. There is only one flaw in your logic. As they worked for 16 days at 50% efficiency, it would have taken extra 16 days to finish job (ONLY IF THEY WORKED AT 50% efficiency). But as they start working at 100% after those 16 days, they are in the deficit of only 16/2 = 8 days. As 4 extra days are provided, the answer is 4. What do you think?

Yes, I see the flaw in my reasoning Clearly, 4 should be it.

In the first place, I don't know how the company came up with the idea of just needing one day extra if it rained for 4 days.

You see, 80 days = 100% of job

For 4 days, you complete 5% of the job, and that's on a good day. If it rains, you only get 2.5% because of 50% drop in efficiency
Now you have 76 days remaining to complete the job, that translates to 95% of the job
So total job completed = 97.5%. You need to complete another 2.5%. This is equivalent to (2.5/100)*80days = 2 days more (assuming on a good day) and 4 days more (assuming rainy days)

So the company should be allocating 84 days to complete the job (better to consider worst case). But the question says the company expects it to be done in 81 days ? How can that be?

great problem, thanks to smc for clear explanation