I'm taking a look at your question now.
You seem to have worked out that your husband will qualify for split year treatment for 2015/16, the tax year he left the UK to work full-time abroad. Assuming that is the case and he is non-UK resident in 2016/17, the fact that he sends money back to the UK to pay bills such as the mortgage does not make that cash taxable. That would defeat the object of qualifying for split year treatment. To qualify for split year treatment, you have to be resident in the UK in that tax year. If you aren't UK resident and don't therefore qualify for split year treatment, then by definition, you are non-UK resident and won't pay UK tax on your foreign earnings.
I'm assuming that you have seen Table E on page 53 in RDR3 here as you mentioned 82 days. So long as your husband did not spend more than 82 days in the UK between 22 May 2015 and 5 April 2016 inclusive and he complies with the other criteria, he will not pay UK tax on his foreign earnings. The number of days he can spend in the UK in the tax year of departure from the UK is reduced because of the fact that he left the UK part way through the tax year not by the number of ties he has. As I said earlier in this answer, he has to be treated as UK resident in the tax he left the UK in order to qualify for split year treatment.
I hope this helps but let me know if you have any further questions.
a The ties rules doesn't apply in the tax year of departure from the UK See Table E on page 53 here. Your husband left the UK between 1 May 2015 and 31 May 2015. He was, therefore, allowed to spend 82 days in the UK in 2015/16. There is no mention of ties to the UK from paragraph 5.1 to 5.25 inclusive which cover leaving the UK. If you look above Table F on page 60, the number of UK ties are relevant for the period between 6 April and the date of a return to the UK inclusive. A look at the flowchart for "leavers" here will show that there is no mention of ties to the UK.
1 Look at paragraph 1.9 here. The UK ties only come into play if the automatic overseas tests and the automatic UK tests are failed. Look at the bottom of the flowchart here. The box is labelled "non-conclusive tests".
2 Accommodation is only relevant when he leaves and returns to the UK or if he fails all of the automatic tests for any one tax year. It is irrelevant as far as the automatic overseas tests are concerned.
As a rule of thumb, your husband needs to be spending less than 91 days in the UK in any one full tax year. So long as he is automatically non-UK resident, the UK ties test does not apply.
Clearly, your husband can't prove that he not UK resident in 2016/17 definitively until after the end of that tax year but HMRC will allow a split year claim for 2015/16 in advance. Don't send anything to HMRC unless they ask for it. All you need to do is make sure your husband spends no more than 90 days in the UK in any one tax year.