the average of the mean solar time for a particular time zone

Apparent solar time is ahead of the clock in Spring and Fall. Apparent solar time is seen while using a sundial.

It very much depends on what is meant by solar time.

Not really, it's fairly clear. There are 2 types of solar time:

1) "Apparent solar time". At any particular place this is called the "local apparent solar time", because it depends on longitude.

2) "Mean solar time". This averages out the natural variations of "apparent solar time". (The word "mean" is just a way of saying "average".) This too is depends on longitude of course, but time zones

are used to cover a wide area, for convenience.

Because, when using "apparent solar time", the length of a "solar day" varies slightly during the year. (This is because the Earth's orbit isn't exactly circular and the Earth's axis is tilted.)

"Clock time" is based on an average (or "mean") of these day lengths, called the "mean solar day". So clocks use "mean solar time".

(By coincidence, on the date the question was answered (14th April) "apparent solar time" and 'clock' time are synchronised.)

There's another reason for apparent solar time being different from clock time.

"Clock time" uses time zones. So, over a wide area, the time on a clock equals the "mean solar time" at a particular, defining, longitude.

the distance between the solar time i.e Mean time apparent solar time is stated as equation of time.

M.t-A.t=equation of time