Any ideas how to fix shallow tree roots?

We live in Taupo, and due to the pumice laden soil, tree roots tend to grow outwards in search of water, instead of down, like this: https://trademe.tmcdn.co.nz/photoserver/full/387314183.jpg This means the roots are easy to hit with the lawnmower blades, like this: https://trademe.tmcdn.co.nz/photoserver/full/387314216.jpg

I've raised the mower blade height, but still seem to scalp the roots a little due to contour of the ground. Would simply covering in topsoil work? Does anyone have any ideas how to solve this problem please?

raising the soil a little seems a good idea. for now,
may need repeating later?

bit of boxing, some bark and a park bench, and maybe a couple of pots either side of the bench with some coloured shrubs

Yes, it is a bit of an issue, isn't it. There are a couple of ways to - I won't say remedy it, but at least improve the situation. Depending upon the levels of the existing soils, you could spade back the existing turf and add soil underneath but you want to avoid a situation where you get the soil built up too far close to the trunk. So, do not raise the soil level at the trunk. Another alternative would be to determine an area under the tree, put a margin on it(bricks, timber etc) and backfill with stones or some such medium (after killing or removing the grass) and use it as a place for sitting or maybe massed pots of colour. Depending upon the type of tree you have, you may be setting yourself up for suckering from the damaged roots - which would be a real pain.
The way to alleviate this problem is acknowledge it in the first place - at planting, use a tree guard. However, not cheap, especially when you add it to the already costly exercise of purchasing trees these days. it means a much bigger hole to dig initially and I would recommend the use of a length of novaflow filled with stones planted at the same time as the tree. This will allow you access the roots with water when needed (as opposed to the surface) and the tree guard will guide the roots downwards and not allow them to go sideways (at least for the depth of the treeguard)

I'd dig down around the offending root and simply cut it off as far back to the main trunk as required to have the cut end under the surface

you have 1 bad one at present and another 2 soon to be a nuisance, leave a few yrs between cutting, lets the tree develop replacements
or dig down around those other two and force them down at least 150mm, in pumice the issue will be keeping them down, I think removal will be the only effective answer, or as others suggest camouflage them

cheers guys!

If it's a valuable tree, or cutting out the roots doesn't appeal, you could put edging around the exposed roots, and fill it in with bark or dirt over weedmat. In other words, build a garden bed around the base of the tree so it's not sitting directly on the lawn, and you don't have to mow directly under it.
I've also seen old trees with edging in a large curve around the roots, filled in with bark chip and a garden bench set in the area of bark. Looked pretty good.

I would "axe" out the offending roots - we have done this. The trees just send out new roots going deeper down. Hasn't affected our tree growth at all.

I wouldn't have a clue, but would it be possible to dig under the root and peg it down somehow (put a cut in it so it bends) and then fill it up with dirt.

A tree??

Build a lovely raised semi-circular garden around and over them. Tree will love ya for it!