Depending on the weight of the steel/iron I imagine what you are seeing is surface rust which will probably not compromise the integrity of the furniture.
The problem with powder coating is that it can crack and blister so you end up with water ingress and things can continue to oxidise underneath the coating. Whatever treatment you opt for it is important to get all the loose material off first.
This can be a pain in the ass (especially if the ironwork is decorative with scrolls and mouldings etc.) and where I have had to restore more expensive/elaborate pieces I have got them finely shot blasted to get back to bare metal.
If you are looking to do a touch up job you might be to find that a decent wire brush is sufficient (maybe try some paint stripper to a small area if you have some, though personally I have not had much luck with it on powder coatings).
Painting over existing powder coat can be problematic as some paints can have trouble with adherence unless the powder coat is old and /or well prepared. A circular wire bush in a cordless drill might help to rough it up if it can't be stripped or sanded easily.
I tend to avoid Hammerite because I find it too viscous which can lead to poor coverage on intricate or pitted surfaces (pinholes can appear if you are not careful). You also need to use special Hammerite thinners to clean brushes or thin the paint.
In my opinion you are always better using a decent primer and topcoat rather than a 'one coat' product. I have always had excellent results with Teamac who produce paints for agricultural and marine applications. I use their zinc phosphate primer with their Metalcote enamel as a top coat - one or two coats of each can last for decades. It can be thinned with ordinary white sprit and even if applied by brush it can give a super smooth finish (almost as good as a spray finish). There are also loads of colours available.
If there isn't a nearby stockist (e.g. Brewers) it can generally be found online in the UK.
https://www.teamacagricultural.co.uk/products/category/plant-machinery-enamel/metalcote-machinery-enamelhttps://www.teamacagricultural.co.uk/products/category/primers/zinc-phosphate-metal-primer