Is there a term for an externality in which prices for a good go up due to consumers being subsidized to purchase or consume that good? Is there a term for an externality in which prices for a good go up due to consumers being subsidized to purchase or consume that good?

So for example, let us say that the government covers part of your medical costs, and in response, private clinics raise their prices because they estimate that they can charge what they were before and then-some, proportional to the added subsidy provided by the government.
That's a standard case of a Consumer Price subsidy. Its effect will be that of market expansion: higher quantity at higher equilibrium price, irrespective of whether it will be lump-sum (a fixed monetary amount) or a fixed proportion of the price.  

A) LUMP SUM SUBSIDY
Consider a monopolist. The monopoly faces a downward-sloping demand curve, say

$$q_d = a -b(p-S) = (a+bS) - bp$$

where $S$ is the lump-sum subsidy. Here the demand schedule shifts upwards by $bS$ maintaining the same slope. Profits are (assume linear costs for simplicity)

$$\pi = p\cdot q_d -c\cdot q_d = (p-c)\cdot [(a+bS) - bp]$$

and the first order condition  for a maximum gives

$$(a+bS) - bp -b(p-c) = 0 \implies p^* = \frac {a+ bS +bc}{2b} = p^*|_{S=0} + \frac {1}{2}S$$

Profit-maximizing price is increasing in the lump-sum subsidy. Now substitute this into the demand equation

$$q_d^* = (a+bS) - b\frac {a+ bS +bc}{2b} = \frac {a-bc}{2} + \frac{b}{2}S = q_d^*|_{s=0} + \frac {b}{2}S$$

Quantity demanded also increased. So the lump-sum subsidy leads to market expansion.  

B) PROPORTIONAL SUBSIDY
Here we have 
$$q_d = a -b(1-s)p,\;\;\; 0\leq s&lt;1$$
The demand schedule changes slope (becomes more "flat"), maintaining the same intercept. 

$$\pi = p\cdot q_d -c\cdot q_d = (p-c)\cdot [a -b(1-s)p]$$

and the first order condition  for a maximum gives

$$a -b(1-s)p -b(1-s)(p-c) = 0 \implies p^* = \frac {a+ b(1-s)c}{2b(1-s)} = \frac {a}{2b(1-s)} + (c/2)$$

Equilibrium price is (non-linearly) increasing in the proportional subsidy. Substitute this into the demand schedule to get

$$q_d^* = a -b(1-s)\frac {a+ b(1-s)c}{2b(1-s)} = a -\frac {a+ b(1-s)c}{2} = \frac {a-bc}{2} + \frac {bc}{2}s$$

Here too equilibrium quantity is increasing in the subsidy.  

For a sector like Health services, where increased market access is of concern (in some countries) due to societal values (and/or cold calculations as regards the material benefits to society in having more healthy members), and where "quantity demanded" is not based so much on desires but rather on not-negotiable needs (illnesses, accidents, etc), the rationale for such subsidies is that the quantity increased will mainly come from new "consumers", previously excluded from the market due to the pre-subsidy equilibrium price in combination with their income level.