This paper proposes a quasi-asymptotic-preserving hybrid discontinuous Galerkin (HDG-QAP) scheme for the resolution of highly anisotropic diffusion problems. The HDG-QAP scheme introduces an auxiliary unknown which serves to capture the information on the dominant diffusion scale. We show that it is well posed for any $\varepsilon>0$, with $\varepsilon$ being a small constant and publications-scientifiques-360/an-iterated-quasi-asymptotic-preserving-hybrid-discontinuous-galerkin-method-for-highly-anisotropic-diffusion-problems-66126.htm/\varepsilon$ characterizing the anisotropy strength, and that its solution is bounded uniformly in $\varepsilon$. At this point, the standard HDG procedure, namely the static condensation on the numerical trace, turns out to violate these uniform bounds. Instead, we show that the static condensation on the auxiliary unknown does lead to similar bounds uniform in $\varepsilon$, but its resolution is costly in terms of computational efforts. Therefore, we propose a relaxation method, called the HDG-QAP Uzawa iteration, to overcome this challenge in which each iteration is fast to compute. We show that the HDG-QAP Uzawa iteration converges for any $\varepsilon>0$, but also for $\varepsilon=0$. Finally, we provide some numerical examples to confirm the findings, and in particular to show that the proposed HDG-QAP Uzawa iteration works well even with a severe anisotropy $\varepsilon=10^{-15}$ where the quality of the numerical solutions is unaffected by the anisotropy strength.