That is kind of a difficult sentence to parse. Does this retain the meaning?

API is used to schedule a one-time call to an arbitrary function at a specific
future time.

The original is an "aside":

Consumers of this API are required to allocate a callout structure (struct callout) for each pending function invocation.

The parentheses are probably the wrong markup for the new form, and "struct callout structure" is redundant. Maybe this:

Consumers of this API are required to allocate a
.Ft struct callout
for each pending function invocation.

As above, use .Ft and the word "structure" is redundant.

"should be" is a recommendation. Does it mean I ought to call those functions before freeing, but will work fine without it? Otherwise, consider "must be drained by a call to".

The rule is to use American English spellings unless the entire document was written with British spellings. See
https://www.freebsd.org/doc/en_US.ISO8859-1/books/fdp-primer/writing-style-guidelines.html

It would be better to say
.Pq Deprecated, see blahblah for current usage.
This function is used to prepare a

Use .Ft and remove the redundant word "structure".

"Was" is a little odd here, but "were" is not quite right either. How about:

function will return as if no timeout was pending.

Again, better to say the function is deprecated and give a pointer to the replacement in a separate sentence. Then explain what this one does in the next sentence.

.Ft and s/structure//

.Ft and s/structure//

Possessive: application's

Comma:
functions, including the

s/Else/Otherwise,/

Avoid simultaneous access by obtaining an exclusive lock

Split this long sentence:

function is called.
It is also expected, but currently not

This last part needs to be split out into a separate sentence, starting with ", except when the".

.Ft and s/structure//

Passive "should be" and should, and "specify a pointer"... How about:

argument is a non-zero pointer to a valid

The specified mutex is expected to be locked when calling any

functions other than the

functions.

The callout function is assumed to have released the specified

s/Else/Otherwise,/

This function is similar to the

function, but accepts a read-mostly type of lock.

s/initialised/initialized/

This function is similar to the

function, but accepts a reader-writer type of lock.

Should that be "read/write" instead?

As above, separate sentence to say it is deprecated and point to the replacement.

argument is a valid pointer to a function that takes a single

"canceled" is spelled "cancelled" elsewhere in this man page. The first is modern American.

s/Else/Otherwise,/

s/called/called,/

s/words/words,/

s/Else/Otherwise,/

s/Else/Otherwise,/

s/Else/Otherwise,/

.Fa pr
argument.

is redundant, and can be just

.Fa pr .

(There are lots of these.)

Avoid the informal use of "you":

This function is used when high precision timeouts are needed.

Again, redundant. Can be just

If
.Fa sbt
specifies...

"the" is not needed here.

Again, can just be

By default,
.Fa sbt
is treated...

Also, "is treated *as*".

Needs a serial comma before "and":
...1/4, and so on.

This function works like

.Fn callout_reset_sbt ,

except the callback function given by

will be executed on the CPU which called this function.

Otherwise, the callback function given by

will be executed on the same CPU...

The rest of that line is unclear. As shown previously in the man page? As run previously on the CPU?

This function works like
.Fn callout_reset_sbt ,
except the callback function given by
.Fa func
will be executed on the CPU given by
.Fa cpu .

Refer to
.Fa callout_init .

Missing the word "if".

British->American: s/behaviour/behavior/

Add comma after "zero".

Use "cannot". "can not" with a space means "not doing it is allowed".

s/timeouts./timeouts are needed./

s/by/by the/

s/CPU/the CPU/

If this means "CPU selection cannot be changed while the callout subsystem is processing the callback for completion.", then s/can not/cannot/.

s/by the/by/

Either "given by the func argument will" or "given by func will". The second is less wordy,