The problem is that you are equating too many things to $\dot{q_k}$.
Usually $\dot{q_k} = \frac{dq_k}{dt}$, a total derivative, as opposed to a partial derivative.
If $q_k$ has no explicit time-dependence, i.e. it does not depend directly on $t$ itself, then $\frac{\partial q_k}{\partial t} = 0.$
In this case, the Poisson bracket reduces to: $ \frac{dq_k}{dt} = \frac{\partial H}{\partial p_k} $, so now you can say $\dot{q_k} = \frac{\partial H}{\partial p_k} $.
If you now consider your $q_k$ to have a time dependence, so $q_k(t)$, the Poisson bracket becomes, as you have pointed out: $ \frac{dq_k}{dt} = \frac{\partial H}{\partial p_k}+\frac{\partial q_k}{\partial t}$, so $ \dot{q_k} =\frac{\partial H}{\partial p_k} +\frac{\partial q_k}{\partial t} $.
ADDITION after discussion in comments:
$\frac{d}{dt}f$ is a total derivative with respect to time, which means it picks out ALL time dependences of $f(t, x(t), y(t), z(t))$. Collectively, call $x,y,z$ as $\{x_i\}$.
Using the chain rule, $$ \frac{d}{dt} = \frac{\partial}{\partial x_i}\frac{\partial x_i}{\partial t} = \frac{\partial}{\partial x}\frac{\partial x}{\partial t}+\frac{\partial}{\partial y}\frac{\partial y}{\partial t}+\frac{\partial}{\partial z}\frac{\partial z}{\partial t}+\frac{\partial}{\partial t}\underbrace{\frac{\partial t}{\partial t}}_1 = \mathbf{u}\cdot\mathbf{\nabla}+\frac{\partial}{\partial t}.$$
In Hamiltonian mechanics, you parameterise a function in terms of its positions $\{q_i\}$ and its momenta $\{p_i\}$. Basically the $\{x_i\}$ from earlier are now $\{q_i, p_i\}$.
$$ \frac{df}{dt}=\frac{\partial f}{\partial x_i}\frac{\partial x_i}{\partial t} = \frac{\partial f}{\partial q_i}\frac{\partial q_i}{\partial t}+\frac{\partial f}{\partial p_i}\frac{\partial p_i}{\partial t} + \frac{\partial f}{\partial t}.$$
The definition of the Poisson bracket is:
$$ \{f,H\}=\left(\frac{\partial f}{\partial q_i}\frac{\partial H}{\partial p_i}-\frac{\partial f}{\partial p_i}\frac{\partial H}{\partial q_i} \right). $$
Plugging in the Hamilton's equations:
$$ \{f,H\}=\left(\frac{\partial f}{\partial q_i}\frac{dq_i}{dt}+\frac{\partial f}{\partial p_i}\frac{dp_i}{dt} \right), $$ when $q_i$ and $p_i$ are the fundamental variables, so they are not functions of $x, y, z$ but only time. This means that, applying the formula from before, $\frac{\partial q_i}{\partial t} = \frac{dq_i}{dt}$, so:
$$ \{f,H\}=\left(\frac{\partial f}{\partial q_i}\frac{\partial q_i}{\partial t}+\frac{\partial f}{\partial p_i}\frac{\partial p_i}{\partial t} \right). $$
As you can see $\frac{\partial f}{\partial t}$ does not enter the definition.
In conclusion,
$$ \frac{df}{dt}=\{f,H\}+\frac{\partial f}{\partial t}. $$
P.S. I have been using Einstein's summation convention: repeated indeces imply summation. $$\frac{\partial f}{\partial p_i}\frac{\partial p_i}{\partial t} = \sum_{i=1}^{N} \frac{\partial f}{\partial p_i}\frac{\partial p_i}{\partial t}.$$
Best Answer
You are confusing in the index, such calculations must be carried out very carefully. I would start with your difention. $$M_i=\epsilon _{ijk} q_j p_k$$
$$M_p=\epsilon _{pnm} q_n p_m$$ $$\{M_i, M_p\}=\sum_l \left(\frac{\partial M_i}{\partial q_l}\frac{\partial M_p}{\partial p_l}-\frac{\partial M_i}{\partial p_l}\frac{\partial M_p}{\partial q_l}\right)$$
First term
$=\epsilon _{ijk}p_k\delta_{jl}\epsilon _{pnm}q_n\delta_{ml}=\epsilon _{ilk}p_k\epsilon _{pnl}q_n=(-1)\epsilon _{lik}p_k(-1)^2\epsilon _{lpn}q_n=-\epsilon _{lik}p_k\epsilon _{lpn}q_n=-\left(\delta_{ip}\delta_{kn}-\delta_{in}\delta_{kp}\right)p_kq_n$
Here I used the antisymmetry of $\epsilon _{lik}$ and equation $\epsilon_{ijk}\epsilon_{imn} = \delta_{jm}\delta_{kn} - \delta_{jn}\delta_{km}$
Second term
Absolutely the same calculations. $=\epsilon _{ijk}q_j\delta_{kl}\epsilon _{pnm}p_m\delta_{nl}=\epsilon _{ijl}q_j\epsilon _{plm}p_m=\epsilon _{plm}p_m\epsilon _{ijl}q_j=-\epsilon _{lpm}p_m\epsilon _{lij}q_j=-\left(\delta_{pi}\delta_{mj}-\delta_{pj}\delta_{mi}\right)p_mq_j=$
Make the change $m=k,j=n$. Then
$=-\left(\delta_{pi}\delta_{kn}-\delta_{pn}\delta_{ki}\right)p_kq_n$
All together
$\{M_i, M_p\}=-\left(\delta_{ip}\delta_{kn}-\delta_{in}\delta_{kp}\right)p_kq_n+\left(\delta_{pi}\delta_{kn}-\delta_{pn}\delta_{ki}\right)p_kq_n=\delta_{in}\delta_{kp}p_kq_n-\delta_{pn}\delta_{ki}p_kq_n=p_pq_i-p_iq_p=q_ip_p-p_iq_p$