I am planning on going to the US for a month to visit a friend who lives there. The main purpose of this trip is for a vacation. However, my employer (UK based) are happy for me to work remotely from there for a few weeks, so I can extend my stay.
Will an ESTA cover me for this, or will I need another specific visa?
Technically there is no visa for the US that will allow you to work remotely while visiting. Here is a nice summary of all the potential visas and what is and isn't allowed. While there are some visas that do allow work, you are unlikely to be able to get them without some serious immigration lawyer like effort. And such effort would be overkill for a simple 1 month visit even if you were eligible for such a visa.
With that said, carrying a laptop with you while you travel is no longer an unusual activity. As long as you don't mention anything at all related to work when entering the country, US immigration shouldn't refuse your entry. But of course, never lie to immigration authorities. They've seen it all and can spot lies from a long way away.
I'd also like to point out another question on here that explores what the nature of work actually is in terms of the B1 visa:
Emails and conference calls while in the USA under ESTA (B-1 Waiver)
Let me share my case. Maybe it will help.
I used to spend a year in total in US business trips on B1 visa during 2011-2013. I confirmed at embassy interview and on the border control I will continue work for my company while in US and continue to get salary. I'll attend meeting and will consult with US customer about our company's system. I worked 40h a week par of that time was meetings and part was work for my non-US company (bug-fixing, performance optimization, code-review, etc). Not sure it was important but my company was covering my expenses and provided a cover letter.
I believe B1/B2 business visa covers such activity in case:
