OP I would be sanctioning for the lying.
How to get him to do the homework....when ds got home from school, he was allowed 30 minutes to grab some fruit/toast and a drink and chat about his day. He then started his homework, and had to have it finished by the time dh got home from work, and I had cooked dinner. If he got it done before that, fine, but I had to see it and check it was OK. Homework was done at the table without distractions.
I also made sure I had a copy of his homework timetable, so I knew what he should be doing (I have been a Year 7, 9, 10 and 11 form tutor, so know the homework dodges). If I was happy with the quality of the work, he could get himself sorted for the next day, and then his time was his. If I wasn't, he redid it until I was. If he got stuck, I helped if asked
Ds had been to prep school before we moved abroad, so it was just a case of putting the same homework structure in place, and sticking to it on the nights when he wasn't doing scouts or swimming. If you can get them in good habits before Year 10 it helps enormously.