I think it depends on how set in stone your budget is. Last time I looked at horses that were around £1K more than my budget but it wasn't totally fixed and I was prepared to be a bit flexible for the right horse. As it was, the horse I bought was £2K lower than my budget and they accepted an offer that was £500 below the asking price.
I always look up to £1k over budget, higher if budget is over £10k, that way you will not miss one that is £250 over budget or one that drops in price into your budget after being on the market a while, most people build in some wriggle room, 10% at least, getting tack included is useful for a first time buyer but it should be a bargaining point and not expected unless it is stated in the ad.
I usually try and sell with tack, basic rugs if the horse is genuinely outgrown/ owner giving up but if they are moving onto something similar in size they may want to keep most of the stuff in case it fits the new one, so don't let not coming with tack be a deal breaker even if it means a bit of haggling to get something within budget, 5k is reasonable so there should be a few to look at.
I would suggest to give about 10% leeway but I doubt you will get many people willing to negotiate much if including tack and rugs - more likely that you might get them to throw in tack and rugs if you offer full asking price
I have little experience buying in the UK, but here in Ireland you'd be thought mad if you didn't haggle. Generally at least 10% leeway, but I have bought horses for less than a third of their asking price ;-)
Went to look at a horse 5k, rode it, found out it could be difficult and offered 1k. Offer was accepted and horse never put a foot wrong!! This was ages ago,not sure I would have the nerve to do it now!
I would factor things in like how long it's been on the market and if it's straight forward. I expect as this is your first horse you are looking for something straight forward so there is likely to be less wriggle room with the price.
I've bought several 3k horses for 1.5k but they were over priced for what they were, not everyone's cup of tea, and I could be seen to be a competent and cash purchaser so got a good deal.
Best horse we bought was 8yo 16.3 ID hunter. Advertised for £4500. Tried him in their field and he was awful hacked him out and he was super. Drove home rang up and offered £2500 increased to £2750 cash no vet. Picked him up the following weekend. Still got him 10 years later and he hunted today.
I generally don't expect that its open to offers unless it states that. I do get a little annoyed when people offer me less (especially a lot less, just rude). I think a few hundred quid maybe is acceptable but I wouldnt be knocking off thousands. If I saw an advert and really wanted to see the horse, I'd contact the seller and tell them upfront what my budget was and if they'd even consider it. If so, I'd go see it and if not, I wouldn't. Then you're not wasting anyones time.
On the other hand I've sold some of my daughters ponies had so much interest I've not dropped a penny in price. Was pretty miffed on one of them first viewer I spent 3 hours with and they had £1000 less than asking price to spend!! I had a queue of people waiting to see her. The second viewer bought her for full asking price no quibbles.
If you're more than about 500 under and really couldn't stretch to the asking price if it is perfect then imo it's very rude not to let the seller know in advance to see if they are still open to you viewing.
That said, Ihave been to see some that I could afford but realistically were so overpriced that unless the horse at viewing was completely different to what was advertised there was no way i would pay more than 3/4 of asking price.
So realistically if I look at around 6k, be prepared to haggle down a little 500-1000 or to include tack, if needed and the horse was right I could always see if they'll hold on a deposit until I have the difference, but I think offering 5k for a horse valued at 6k isnt as bad as offering say 2k for one up for 3k?
I always went into it thinking haggle. They can only say no and you can always up the price if they say no to your first offer if you really like the horse. I think I ended up paying about £200 more than the asking price but I got them to include an almost new saddle and a bridle with it, so cheaper then buying the horse at asking price and buying saddle and bridle after.
A friend of mine was pony hunting a few years back, had a relatively small budget and had two to view in a weekend, one was very slightly over-budget and twice the price of the other. Went to see the cheaper one first, the owner was describing him and said "that's why he's so cheap" to which my friend agreed he was cheap and said "well I agree he's cheap so won't be haggling with you about the price" only to realise when she got back to the car that she'd got mixed up and this was the more expensive pony!
Lesson - double check the advert/details before you get out of the car!!!
He was actually a fab pony who could turn his hoof to anything, and be ridden by anyone, her kids enjoyed him for years and he turned out to be an absolute bargain, pony of a lifetime type, now retired.