Circle above a letter

Not exactly the answer to your question, but the package yhmath have the widering command which can be useful to you:

documentclass[twoside]{article}
usepackage{fancyhdr}
usepackage{nameref}
usepackage{abraces}
usepackage{amsfonts,amsmath, amssymb, mathtools}
usepackage{yhmath}
makeatletter
newcommand*{currentname}{@currentlabelname}
makeatother

fancypagestyle{yourStyle}{
fancyhf{}
fancyhead[RO,LE]{thepage}
fancyhead[RE]{leftmark}
fancyhead[LO]{MakeUppercasecurrentname}
 }

begin{document}

 $$left((overline{M})_{tau_1}right)^mathsf{c} =  (aoverbrace[L1R] 
 {M^mathsf{c}}^{circ})_{tau_1}$$
 $$left((overline{M})_{tau_2}right)^mathsf{c} = (aoverbrace[L1R] 
 {M^mathsf{c}}^{circ})_{tau_2}$$

 $$left((overline{M})_{tau_1}right)^mathsf{c} =  (widering{M^mathsf{c}})_{tau_1}$$
 $$left((overline{M})_{tau_2}right)^mathsf{c} = (widering{M^mathsf{c}})_{tau_2}$$

end{document}

I have changed the initial code completely: there are the combination of two particular packages: accents and scalerel. With these packages I have created a macro called arcsup with the arc only above the character . If you want to have a large horizontal arc you can increase the value of hstretch{1.8} to stretch horizzontally the arc.

documentclass[a4paper,12pt]{article}
usepackage{amssymb,mathtools}
usepackage{accents}
usepackage{scalerel}
newcommandarcsup[1]{accentset{overset{scriptscriptstylecirc}{hstretch{1.8}{frown}}}{#1}}
begin{document}

[bigl((overline{M})_{tau_1}bigr)^{!mathsf{c}} =  (arcsup{M}^mathsf{c})_{tau_1}]
 end{document}

You can see the increment to

hstretch{2.5}

Addendum: If you prefer, a smaller circ symbol you can use the option,

scaleto{circ}{2pt}%%%% <----- 2pt, or any value neq 0

---

documentclass[a4paper,12pt]{article}
usepackage{amssymb,mathtools}
usepackage{accents}
usepackage{scalerel}

newcommandarcsup[1]{accentset{overset{scaleto{circ}{2pt}}{hstretch{1.8}{frown}}}{#1}}
begin{document}

[bigl((overline{M})_{tau_1}bigr)^{!mathsf{c}} =  (arcsup{M}^mathsf{c})_{tau_1}]
 end{document}

these are the screeshots for comparison:

scaleto{circ}{1.5pt}

In the recent version 1.18 of newtxmath there are new math accents and macros. The macro widering, as written into the guide, places a ring centered over an overgroup, not dissimilar from its use in yhmath. But for my tastes the command widering given by newtxmath have a better arc:

documentclass[a4paper,12pt]{article}
usepackage{newtxmath,newtxtext}

begin{document}
[bigl((overline{M})_{tau_1}bigr)^{!mathsf{c}} =  (widering{M}{}^{mathsf{c}})_{tau_1}]
end{document}

See this output:

Or changed the position of superscript:

documentclass[a4paper,12pt]{article}

usepackage{newtxmath,newtxtext}

begin{document}
[bigl((overline{M})_{tau_1}bigr)^{!mathsf{c}} =  (widering{M^{mathsf{c}}})_{tau_1}]
end{document}