I have a question on when I can use an intermediate variable from which function value, jacobian and (projected) hessian are directly derived during optimization.

For example, I have a function f depending on q, which in turn depends on x.
A good property of f is that its jacobian g and projected hessian hv all depend on q directly.
Therefore, I can compute q from x once and obtain all f, g and hv in the meantime.
In python/scipy, I can set q to be a global variable. In each optimization iteraction, in the evaluation of f with myfunc, q is automatically updated based on the current x; in the evaluation of g and hv, instead of dealing with x, the myjac and myhessp functions read the global q and return the corresponding derivatives.
Something like

q=0

def myfunc(x):
    global q
    q=foo(x)
    return bar0(q)

def myjac(x):
    # do nothing with x
    return bar1(q)

def myhessp(x):
    # do nothing with x
    return bar2(q)
scipy.optimize.minimize(myfunc,initial_guess_x0,method="blablabla",jac=myjac,hessp=myhessp)

In each iteration, if myfunc, myjac and myhessp are evaluated only once respectively with only one $x$, the strategy will definitely work.
However, some optimizers evaluate these values with more than one x in one iteration.
When the optimizers try to evaluate g and hv with a x_prime other than the current x that is used to evaluate f, the x_prime actually does not make a difference and instead the q derived from x is read by mistake.
Of course, multiple evaluation of f is tolerable, since in myfunc, f is obtained after q is obtained from x.

One example is the use of a finite difference numerical hessian.
To evaluate the numerical hessian, the optimizer will need to call myjac with many different xs.
Since myjac only reads the unchanged q instead of x, all the jacobian will be exactly the same.

I have tried letting myjac and myhessp calculate g and hv from x instead of q, but in my case the evaluation of q is very time-consuming.
In contrast, obtaining g and hv from q is quite fast.
Therefore, I want to stick to the strategy, and I wonder if any optimizer works with it.
The optimizer had better be a second-order one, so the convergence will be easy.
It had also better be based on a level-shifted algorithm with trust radius, because I have a good preconditioner at hand.

Many thanks in advance!