Fixed gear bike chain alignment I have recently converted my old early 90s 18 speed Peugeot road bike to a fixed gear.
The chain is clicking when pedalling and is out of line by about 10mm as you can see in the photo.
The rear cog is a new 18t fixed gear. The chainring on the front is the from the original bike is 52t and was the 3rd gear previously. I have removed the two other chain rings.  The new chain itself is 1/8th inch.
If I get a new square tapered crank set and chain ring is this likely to improve the alignment? Will this bring the chain ring closer to the bike and more in line with the rear cog?
The pic makes it unclear but I assume flopping the ring in to the middle position has been done and not helped.
First you need to decide whether 52-18 is what you want. It probably isn't. Your answer may inform which route is more economical, since buying a new ring will cost some notable percent of what a new crank would cost. Also, figure out whether the 52 will work in the chainline position you need it to. It very well may not.
If you want to make this crank work, measure your rear chainline and your front chainline. If you have a symmetrical BB spindle, multiply the difference by two and buy a new BB with a spindle that much longer. If asymmetrical, take its measurements and do the chainline/spacer math to arrive at the spindle length you need to simultaneously get the front chainline matched to the rear while also centering the cranks. The usual offset is 3mm but it's not universal.
If this a modern off the shelf track hub, if you drop in any modern track crank with its recommended spindle length it will basically work in terms of chainline. Meaningful exceptions to that are rare. As before, what may not work is frame clearance. You will have to get a sense of that up front before making your other decisions. In some cases that will drive gear size choices.
I'm pretty sure this won't have a French shell, but you should corroborate that before buying things.