I'm not sure what the patio/path is made of, but I'll assume it is solid, perhaps steel-reinforced.
Some suggestions:
1. After the rain/hosing has stopped, brush the water away with a broom or pool/window-cleaning squeegee.
2. Cut a groove with an angle grinder. Use a board for guidance; it'll look rubbish unless it is straight and clean. The groove will need to be kept clean, of course. If you have the space to wield the angle grinder, consider putting that groove right at the edge. There it will be less obvious, and could lead the water to the soil underneath - so it might drain downwards a little (depending on your clay), as well as sideways.
3. Drill a drain hole through the path/patio near the lowest point, using a masonry tank cutter (as you'd use to put an vent tube through a kitchen/utility room wall for a tumble drier, though it could be smaller than that). ...And continue drilling downwards in the clay below (you can get extension pieces for these tank cutters). Insert a plastic pipe that you have drilled lots of small holes in, and perhaps fill it with gravel (depends on the particle size distribution of your clay). Fit a drain cap. That's a small soakaway. The first thing to do would be to find a drain cap, tube and cutter of compatible sizes.
4. Stand some pots in a line (or perhaps one long trough), with a strip of capillary matting underneath, leading to a lower point just past the path/patio. Asssuming the cap.mat remains hydrophillic, and assuming the deepest point of your puddle is only a few cm, the cap.mat should 'wick' the moisture away. Allow other drainage for the pots/trough as well.
5. Use a concrete floor grinder to wear away some of the patio/path. That wouldn't be feasible if the patio/path has a decorative top which would be worn through. In effect it would be a wider version of the groove in the 'angle grinder' suggestion.
6. Depending on access and on the scale of the patio/path, excavate a little under the high edge, and with the clay thoroughly wet (eg. in winter), put weight (eg. lots of concrete blocks) on the high side and be patient. If such patience is not available (!), and if you have access to a JCB, you can use the bucket to press down gently on the high edge (with old wooden boards between bucket and patio/path). The trick is to avoid the low side rising (cantilever effect) - so you need to press in the middle and ensure the boards spread the load well. Do not rush it else the patio/path may crack.
Let us know what you do and how well it works!