Does the HSE recognise electronic signatures as evidense or do they prefer handwritten with ink

Yes but I presume this depends how you define an electronic signature?

I think an email from someone confirming they have read or checked something is OK, as would an online declaration from someone on a system such as Assessnet be OK - as long as the a person has not stupidly shared his/her password which sometimes happens.

However someone copying and pasting in a scanned signature on behalf of their boss who has not even bothered to read the document is clearly not good.

I must say when it comes to sending off investigation documents to our insurers I still make sure all risk assessments are scanned with proper signatures on them. Even if the HSE would condone this, I would not be happy at the thought of us being open to accusations of poor document control if an injury goes to court. Just sending in un-signed or fake signatures leaves me nervous!

If you mean that this is a proper electronic signature application, not just a copy of a written signature that should be ok. The courts have long accepted these are real signatures, in fact they might be better since most people only apply a scrawl as their signature which could be anything written by anyone.