If you havn't already done so, analyse your bank statements so you know EXACTLY where every penny is being spent, for both yourself and your partner. Go back at least three months, preferably six months.
Until you have a clear view, you'll struggle to make changes.
It's often lots of relatively small/regular things that really add up, such as morning coffees, lunchtime meal deals, etc that simply don't register with you. I'd also wager that your partner is spendthrift with "daytime" spending, such as meal deals (not using a clubcard for discounts or choosing from the meal deal options), or buying pies/bacon buns every morning, a drink at the pub on the way home, etc. - All "invisibles" unless you see their bank statements too!
Then look at other "essentials" like car leases, mobile phone contracts, gym memberships, Spotify/Netflix, Sky/Vrigin TV, home broadband, home electric and gas, magazine subscriptions, etc etc. It's easy just to let the standing orders and direct debits continue when you don't use them anymore, easy to keep a "top" Sky/Virgin TV subscription when you could do with a more basic subscription (or none at all). Always shop around for insurance every year.
Sell anything around the house you don't need/use anymore - kids' toys, kid's clothes, books, CDs and DVDs, sell old furniture and white goods you don't need anymore rather than scrapping them. We keep selling "bundles" of things like books, CDs and DVDs on Ebay - can usually clear £5/£10 after postage for small bundles of several items. We also sell any of our own clothes that are still in top condition but that we've grown out of, i.e. OH lost a few stone in weight, and we sold all his old suits, work shirts, works trousers, etc, casual pants/shorts, etc - bigger sizes are in high demand apparently, even for used stuff - must have cleared a few hundred pounds by selling it all on Ebay that would have otherwise just gone to the charity shop!