\hspace{1pt} is equivalent to what spacing commands, in math-mode?

I would not add space at the points you indicate, so simply not adding space would be my recommendation, however if you add

\showthe\dimexpr 1em /18\relax

In the text of your document you will get

> 0.60834pt.

so 1mu math space is .6pt, a thin space, \, is a space of \thinmuskip which is 3mu in the standard (and most other) classes so 1.8pt at \normalsize. So perhaps you want \mskip2mu

You are specifically looking at spacing in math. For that, a common spacing measure would be mus. With this in mind, here's an extract from TeX by Topic (section 23.6 Mathematical spacing: mu glue):

Spacing around mathematical objects is measured in mu units. A mu is 1/18th part of \fontdimen6 of the font in family 2 in the current style, the "quad" value of the symbol font.

Under the different document class defaults, 1mu therefore is equivalent to

• 10pt: 1em = 10.00002pt; 1mu = 0.55554pt
• 11pt: 1em = 10.95003pt; 1mu = 0.60832pt
• 12pt: 1em = 11.74988pt; 1mu = 0.66666pt

So, for a (roughly) 1pt space, you'd need (roughly) 2mus under any document class font option. These spaces would scale with the font size, so you could define

\newcommand*{\psp}{\hspace{.1\dimexpr1em}}

or

\newcommand*{\psp}{\mskip2mu}