For this math font and combination of letters, I think you need to insert a negative thinspace, \!, i.e., write \alpha_{\!p} instead of just \alpha_!p. This applies to cases where \alpha is on the baseline or in the first subscript position.
The following screenshot shows the letter combinations with this adjustment on the left, and without the adjustment on the right.
Note that, depending on the font in use, there will always be some combinations of letters for which manual spacing adjustments are needed in math mode in order to achieve the optimal typographic look.

\documentclass{scrartcl}
\usepackage{newtxtext,newtxmath}
\begin{document}
$\alpha_{\!p}$ $\alpha_p$
$\lambda_{\alpha_{\!p}}$ $\lambda_{\alpha_p}$
\end{document}
Addendum: If you choose to work with the MathTime Professional II math font package instead of the newtxmath package, you can get the subscript positioning correction automatically by specifying the option subscriptcorrection. (Both mtpro2 and newtxmath are Times Roman Math fonts.) Note that whereas the full mtpro2 package is not free of charge, its "lite" subset -- which is all that's needed for this example -- is in fact free of charge. Observe in the screenshot below that hardly any difference is visible between the version with the explicit \! correction on the left and the automatic correction on the right. (Well, there's a tiny difference, as my manual correction amounts to \mkern-3mu, whereas the automatic one by the mtpro2 package amounts to \mkern-2mu.)

\documentclass{article}
\usepackage{newtxtext}
\usepackage[lite,subscriptcorrection]{mtpro2}
\pagestyle{empty}
\begin{document}
$\alpha_{\!p}$ $\alpha_p$
$\lambda_{\alpha_{\!p}}$ $\lambda_{\alpha_p}$
\end{document}
newtxmathpackage would offer an option calledsubscriptcorrection, similar to the one already offered by themtpro2package. (My answer -- see below -- shows how to use themtpro2package with thesubscriptcorrectionoption.) See this answer by @egreg for the code that implements the subscript correction for any math font package. Good news: Asnewtxmathandmtpro2are both Times Roman math clones, you should be able to use Enrico's code "out of the box", i.e., without needing to modify any of the spacing corrections. – Mico Aug 25 '15 at 13:35