For context, I am a single parent and my income is around the same as you're projecting for myself and 2DC. £1200 take home salary (I can only work part time due to care needs of youngest DC, there's only one childminder who covers their school, and she won't take them any more. Moving schools not an option due to the autism provision within their current school), and around £800 UC support.
We manage because I watch every penny that I spend. It covers bills, DS does hockey (£3 per week) and drumming lessons (paid for by the local council due to pupil premium), DD goes to choir (£21 a month). We have a couple of UK overnight trips a year. I manage the extras because I coupon, buy things when they are on offer, meal plan and batch cook where I can, utilise Too Good To Go and the Community Supermarket to keep our food shopping to around £200 a month, even accounting for DS's ARFID.
I scour social media in the run up to half terms and we utilise as many of the free activities as possible as soon as bookings open, we're also lucky that every Saturday there's a library in walking distance with a free Lego club, so we do that and get new books every week.
I have apps on my phone and the family computer that give us points for playing games which I trade in for gift vouchers, as well as running all of my online purchases through Top Cashback, which also gives vouchers; these cover a significant chunk of Christmas/birthday gifts, as well as pay for tickets to Disney on Ice every year, meaning that trip only costs me our train fair, for which I have a rail card, which again I bought when it was on sale for a 3 year card.
I keep an eye on sales and buy the next size up of clothes for DC as and when I see them cheaply and stash them in the cupboard for when they are needed.
It's doable, but there is no room for impulse buys, I need to plan every penny, and wring as much out of my money as I can. When unexpected expenses happen, it can be a challenge depending on what the expense is, as I have no savings. I could have a small amount to be fair, but I choose to prioritise the trips at summer and Easter, as my DC will only be little once. I have a reasonable credit score, so I can get appliances on interest free finance if needed, and I do have a credit cars with £1000 limit which I only use for the grocery shop then pay off in full to build my credit score further. It was through the floor from debt acquired when I was still with my ex.
It depends on what quality of life you want. Is the constant worry of watching every penny and never being able to spend without shopping around for the best price or finding a coupon going to just be a different kind of stress to your current job?