To try and reduce noise in day of maximum, I tried using a 31 day centred average. Cryosphere today area numbers. Still leaves noisy data, but anyway: The day of maximum looks strangely early in the period 1986 to 1998 and then no trend this century.

Just random noise or is there likely to be an underlying cause, and if so what?

What you are trying to do is find the maximum of a function in the presence of noise, when the function is essentially flat at the maximum. This is a hard problem. Still, I have some thoughts.

Since you didn't say otherwise, I assume you are using an unweighted average. This is still sensitive to the noise at the ends of the averaging window. You may get better results with a weighted average. Depending on what software you are using, you should check for kernel smoothers or binomial smoothers or Gaussian smoothers, or possibly other terms. You may be able to smooth the data better, but with a smaller effective averaging window.

These smoothers all assume the noise is uncorrelated, which is probably not the case. There are statistical tools which perform smoothing with autocorrelated noise, so if you want to go that far you could probably find an R package which will do all the work for you, for example.

Even then, the variance is likely to be large, making year to year comparisons difficult. If you're really motivated, you could do a bootstrap estimate of the variance, but it's probably a lot of effort with little point.

Looking at the graph as is, there's no visual evidence of a trend, but you could do a test for a change point in 1998. Again, depending on what software you are using, you may be able to directly do a statistical test of whether the mean maximum day for years up to 1998 is different than the mean maximum day after 1998.

On the other hand, if there has been a change in the maximum day, I would expect it to be a smooth trend rather than a jump at a change point. Even if the change point test is statistically significant, I'm not sure it would be physically meaningful.