It sounds like you want tick labels, not the tick marks in particular. In most figures, the minor tick marks are already there where you want them.
The following may then work, though I would think there's an easier way. Note that I'm applying labels to the minor tick marks only: the (two) major tick marks already have a label. Unfortunately, the fonts of the two types of tick marks are not the same; I think that's a result of the LaTeX equation usage.
import numpy as np
import matplotlib.pyplot as plt
fig, ax = plt.subplots()
X = np.logspace(0, 3)
Y = X**-0.2
plt.plot(X,Y)
plt.yscale('log')
yticks = np.linspace(-0.1, -0.9, 9)
ax.set_yticks(10**yticks, minor=True)
ax.set_ylim(0.1, 1)
ax.set_yticklabels(['$10^{{{:.1f}}}$'.format(ytick) for ytick in yticks], minor=True)
plt.show()
which results in:

For the issue of the different label fonts, you can manually change the major tick labels:
ax.set_yticks([1, 0.1])
ax.set_yticklabels(['$10^0$', '$10^{-1}$'])
(and probably the same for the x-axis).